妥协

The Compromise

2020年5月，全球焦点集中在新冠疫情，种族正义抗议活动在美国各地爆发之际，一项技术发展在小圈子里引发了兴奋和恐惧。OpenAI公司开发的一个名为GPT-3的计算机程序，生成了一些迄今为止最优秀的计算机生成的模仿人类写作的文字：据作者称，这些虚假新闻文章几乎一半的时间都能骗过人类读者，以及类似华莱士·史蒂文斯风格的诗歌。1该程序基于一个统计模型，该模型的作用是：给定一个单词序列，它会尝试预测接下来会出现哪个单词。接下来。该模型除了使用维基百科和大量书籍的内容外，还使用了从互联网上抓取的超过 570 GB 的压缩文本进行训练。2该系统的创建者将其描述为“任务不可知”学习者，即一种可以执行各种认知任务而无需针对任何特定任务进行微调的机器学习模型。3这种新的人工智能 (AI) 方法旨在改变计算机编程的实践：我们无需设计算法来解决给定的问题，而是用英语告诉机器它的目标，它会计算出（我们希望）正确的答案。

In May 2020, as much of the world was focused on the COVID-19 pandemic and as racial justice protests took place across the United States, a technical development sparked excitement and fear in narrower circles. A computer program called GPT-3, developed by the OpenAI company, produced some of the best computer-generated imitations of human writing yet seen: fake news articles that were, according to the authors, able to fool human readers nearly half the time, and poems in the style of Wallace Stevens.1 The program is based on a statistical model that does one thing: given a sequence of words, it tries to predict what word will come next. The model was trained on more than 570 gigabytes of compressed text scraped from the internet in addition to the contents of Wikipedia and a large number of books.2 The system’s creators describe it as a “task-agnostic” learner—that is, a machine learning model that can perform a wide range of cognitive tasks without having to be fine-tuned for any particular one.3 This new approach to artificial intelligence (AI) aspires to transform the practice of computer programming: instead of designing an algorithm to solve a given problem, one tells the machine its goal in English, and it works out (one hopes) the correct answer.

从人文主义的角度来看，这一说法的一个显著特点在于它如何将知识定位于语言之中。GPT-3 的输入和输出都是文本，而且它只接受文本训练；它没有任何经验，哪怕是在最宽泛的概念意义上。4然而，它表面上的能力并不局限于以不同风格重写段落等面向语言的任务；就其真正的多任务学习者而言，它集写作辅助、可编程计算器和搜索引擎的功能于一身。怀疑地看，这台机器就像一只鹦鹉，说着它无法理解的话。但语言模型可以构成通用方法的基础这一想法，也可能暗示着某种解构性的洞见：学习语言与学习思考无法区分，选择词语的认知操作种类没有限制。如果我们相信研究人员——我们当然不应该不加批判地相信——那么自然语言就是创造难以捉摸的通用人工智能 (AGI) 的基本要素。

From a humanistic standpoint, a striking aspect of this claim is how it locates knowledge in language. GPT-3’s input and output consist of text, and it is trained on nothing but text; it has no experience, even in the loosest notional sense, of anything whatsoever.4 Yet its apparent capabilities are not limited to such language-oriented tasks as rewriting paragraphs in different styles; to the extent that it really is a multitask learner, it unites the functions of writing aid, programmable calculator, and search engine. Skeptically viewed, the machine is acting like a parrot, saying things it cannot understand. But the idea that a language model can form the basis for a universal method could also suggest something like a deconstructive insight: that learning language cannot be distinguished from learning to think, that there is no limit to the sorts of cognitive operations that go into choosing words. If we are to believe the researchers—which we certainly should not do uncritically—then natural language is the essential ingredient needed to create the elusive artificial general intelligence (AGI).

像 GPT-3 这样的大型语言模型的兴起，颠覆了人们长期以来理解的计算与语言关系的范畴。计算机通常被描述为符号操作机器；它们的工作原理是通过不依赖于符号含义的机械规则重新排列电子表示的 1 和 0。GPT-3 重新引发了一场由来已久的哲学争论：这样的机器是否真的可以理解语言？5但即使在这种发展之前，计算机也很少被用作纯粹的、不解释的符号操作器。在现代界面中，屏幕上充斥着诸如“保存”、 “提交”、 “点赞”之类的文字，它们在计算逻辑和人们交流的社会惯例之间起到中介作用。长期以来，工程师们一直将计算机系统的交流元素视为表面的装饰，而数据结构和算法才是计算机程序的真正核心。像 GPT-3 这样的语言模型模糊了这两者之间的界限。由于这些系统完全依赖于有关人们语言实践的数据，它们使得判断算法在哪里结束、语言在哪里开始变得比以往任何时候都更加困难。

The rise of large language models such as GPT-3 has unsettled the categories in which people have long understood the relation of computation to language. Computers are often described as symbol-manipulating machines; they work by rearranging electrically represented ones and zeros through mechanical rules that do not depend on the symbols’ meanings. GPT-3 has rekindled a long-standing philosophical debate over whether such a machine can really be said to understand a language.5 But even before this development, computers have seldom been used as purely uninterpreted symbol manipulators. In modern interfaces, screens are festooned with words—save, submit, like—that serve to mediate between computational logic and the social conventions by which people communicate. Engineers have long treated the communicational elements of computer systems as superficial ornaments when compared to the data structures and algorithms that form the real core of a computer program. Language models such as GPT-3 have blurred the lines. Since these systems depend, through and through, on data about people’s linguistic practices, they make it harder than ever to judge where algorithm ends and language begins.

众所周知，在计算机科学中， “算法”一词的表述比定义更容易。虽然该词最近与机器学习联系在一起，但教科书通常将算法简单地解释为解决问题的精确定义的程序。这些程序通常采用步骤序列的形式，例如以下用于查找书中最长句子长度的算法：

The term algorithm, as it is used in computer science, is notoriously easier to illustrate than to define. While the word has recently become associated with machine learning, textbooks typically explain algorithms, quite simply, as precisely defined procedures for solving problems. These procedures often take the form of sequences of steps, as in the following algorithm for finding the length of the longest sentence in a book:

尽管类似的指令出现在各种各样的环境中——典型的例子是烹饪食谱——但将一个过程称为算法会引发一组更具体的学科实践。编程语言提供了一种以机器所要求的极高精度描述过程的方法。（要使上述过程成为真正的算法，我们必须明确什么是单词和句子——这并非易事。）计算复杂性理论提供了衡量和提高这些过程效率的方法。更广泛地说，算法思维（广义上的算法思维）鼓励抽象。6算法的技术理论鼓励开发可用于不同目的和不同情境的通用解决方案；这些过程被认为是数学实体，它们独立于描述它们的语言的复杂性和使用它们的具体情况而存在。

Although similar instructions occur in a wide range of contexts—a typical example is cooking recipes—calling a procedure an algorithm evokes a more specific set of disciplinary practices. Programming languages provide a way of describing procedures with the extreme precision demanded by machines. (To make the foregoing procedure a true algorithm, we would have to clarify what words and sentences are—not a straightforward matter.) Computational complexity theory provides methods for gauging and improving the efficiency of these procedures. More broadly, algorithmic thinking (in the expansive sense of thinking about algorithms) invites abstraction.6 The technical theory of algorithms encourages the development of general solutions that can be reused for different purposes and in different contexts; the procedures are thought of as mathematical entities that exist apart from the complexities of the languages in which they are described and the concrete situations in which they are used.

本书探讨了这种抽象形式的诞生。它聚焦于算法史前史中的一条线索：符号在数值计算、代数、微积分、逻辑以及最终的计算机科学中的应用。Python 和 R 等标准编程语言借鉴了代数和逻辑的符号表示法（以及其他来源），将其作为精确定义运算的方式。然而，这些符号，如同编程语言一样，长期以来将计算与另一种更难简化为机械规则的功能结合在一起：交流。诸如 F s ₌ kx之类的符号公式既提供了如何计算某些事物的指令（在本例中，是将弹簧拉伸或压缩给定长度所需的力），也提供了一种表达关于世界的命题的方式。7 我认为，现代算法的概念（即计算机科学中的术语）取决于一种将计算与交流的复杂性区分开来的特殊方法，这种方法最早形成于 19 世纪的纯数学中。8尽管机器学习系统经常（令人困惑地）被冠以与算法理论中处理的精确定义的程序相同的名称，但我希望表明机器学习代表着对这种技术概念的突破，这种技术概念将数百年的认识论界限置于危险之中。

This book is about how this form of abstraction came into being. It focuses on one thread in the prehistory of algorithms: the use of symbols in numerical calculation, algebra, calculus, logic, and, eventually, computer science. Standard programming languages such as Python and R draw (among other sources) on the symbolic notations of algebra and logic as ways of precisely defining operations. Yet these notations, like programming languages, have long combined computation with another function that is harder to reduce to mechanical rules: communication. A symbolic formula such as F_s = kx provides both instructions for how to compute something—in this case, the force required to extend or compress a spring by a given length—and a way of conveying a proposition about the world.7 It is my contention that the modern idea of algorithm, as the term is used in computer science, depends on a particular way of disentangling computation from the complexities of communication that first took shape in the pure mathematics of the nineteenth century.8 Although machine learning systems are often called (confusingly) by the same name as the precisely defined procedures dealt with in the theory of algorithms, I hope to show that machine learning represents a break from this technical concept that places centuries-old epistemological boundaries in jeopardy.

算法的历史有长有短。广义上讲，算法思维的历史至少和书面记录一样悠久。9 古巴比伦人在泥板上写下了基于规则的数值计算程序，计算机科学家唐纳德·E·克努斯从中领悟到了他所在学科的雏形。10 算法这个词（早期有多种拼写方式，如 algorism、algorithmus、algram 或 augrym）虽然没有那么古老，但仍然非常古老——它形成于12世纪，源自阿拉伯数学家穆罕默德·伊本·穆萨·花拉子密 (Muḥammad ibn Mūsā al-Khwārizmī) 的名字，他在 9 世纪描述了使用印地语-阿拉伯数字进行计算的技术。11这些技术——包括人们仍然在学校学习的熟悉的加法、减法、乘法和除法程序——构成了最初的“算法”。早在16世纪，“算法”一词就涵盖了一系列超越原始技术的其他技术，通常涉及符号代数。在寻找如今计算领域中那些追求极致的雄心壮志的先驱时，通俗的历史著作通常会将德国博学家戈特弗里德·威廉·莱布尼茨作为例子。从17世纪60年代开始，莱布尼茨尝试创建他所谓的“推理演算器”（calculus reciprocator） ——一种可以解决几乎所有主题争议的符号“计算”系统。科普作家马丁·戴维斯将现代计算机描述为“莱布尼茨梦想”的实现，即将数学符号操作扩展为一种可以应用于任何事物的通用方法。12

The history of algorithms has been told in both long and short versions. In a broad sense, algorithmic thinking goes back at least as long as the written record.9 On clay tablets, the ancient Babylonians wrote down rule-based procedures for numerical computation in which the computer scientist Donald E. Knuth perceived the rudiments of his discipline.10 The word algorithm (early on spelled a range of ways, such as algorism, algorithmus, algram, or augrym) is less ancient but still very old—it was formed in the twelfth century from the name of the Arabic mathematician Muḥammad ibn Mūsā al-Khwārizmī, who described techniques for computing with Hindi–Arabic numerals in the ninth century.11 These techniques—including the familiar addition, subtraction, multiplication, and division procedures one still learns in school—made up the original “algorithm.” As early as the sixteenth century, the word algorithm came to encompass a range of other techniques beyond these original ones, often involving symbolic algebra. In searching for precursors to the totalizing ambitions that now attend computation, popular histories commonly single out the German polymath Gottfried Wilhelm Leibniz. Starting in the 1660s, Leibniz attempted to create what he called a calculus ratiocinator—a system of symbolic “calculation” that could resolve disputes about virtually any topic. The science writer Martin Davis describes the modern computer as a fulfillment of “Leibniz’s Dream” of extending mathematical symbol manipulation into a universal method that can be applied to anything whatsoever.12

迈克尔·马奥尼 (Michael Mahoney) 和洛林·达斯顿 (Lorraine Daston) 等历史学家开展的更有针对性的研究表明，这种笼统的叙述忽视了计算实践在几个世纪以来的变化方式。13马克·普里斯特利 (Mark Priestley) 有力地论证了计算机编程与符号逻辑等其他领域并没有内在联系，而是通过计算机科学家的有意选择而产生的联系。14马修·琼斯 (Matthew L. Jones) 和玛丽亚·罗莎·安托纳扎 (Maria Rosa Antognazza) 将莱布尼茨置于历史背景中，表明他的工作并不完全是现代意义上的算法。15这种更具历史主义色彩的视角导致了一种对比的叙述，其中算法的概念非常新。令人尊敬的算法这个词或许如此，但其含义直到20世纪60年代计算机科学作为一门学科出现后才形成现代形式。《理性如何几乎失去理智》一书的六位作者认为，直到冷战时期，当智库研究人员试图用严格基于规则的决策取代人类判断时，算法才成为理性的典范。16算法成为社会关注点则更为晚近，源于机器学习技术发展与根深蒂固的不平等和歧视结构的交汇。17

More focused scholarship by historians including Michael Mahoney and Lorraine Daston has shown that such sweeping narratives overlook the ways computational practices have changed over the centuries.13 Mark Priestley has argued persuasively that computer programming has no intrinsic relation to other fields such as symbolic logic but rather came to relate to them through intentional choices made by computer scientists.14 Matthew L. Jones and Maria Rosa Antognazza have placed Leibniz into historical context and showed that his work was not exactly algorithmic in the modern sense.15 This more historicist perspective has led to a contrasting narrative in which the concept of algorithm is very new. Venerable as the word algorithm may be, its meaning arguably did not reach its modern form until the 1960s, when computer science emerged as an academic discipline. The six authors of the book How Reason Almost Lost Its Mind have argued that algorithms were not a model of rationality until the Cold War period, when think tank researchers sought to replace human judgment with strictly rule-based decision-making.16 The algorithm’s rise to the status of a social concern is even more recent, stemming from a confluence of technical developments in machine learning with entrenched structures of inequality and discrimination.17

这种对算法概念的历史化解读应该警示我们，不要不加批判地将过去的符号方法与现代算法混为一谈。我们所知的算法是一个复杂的混合体，其史前史涵盖了一系列实践，包括天文和统计计算、官僚程序、市场经济以及诸如美国人口普查之类的政府数据收集工作。作为现代算法的背景，符号方法的重要性与其说在于其内在的相关性，不如说在于它们在技术话语中发挥的作用。在20世纪60年代和70年代，计算机科学学科开始将算法视为抽象的过程，即使以各种方式实现、解释、应用和解读，它们仍保持着稳定的同一性。正如我在本书中所展示的，这种思维方式与一系列关于符号与语言关系的长期争论息息相关。是否应该使用相同的符号来计算结果并将其呈现给他人？符号的含义在多大程度上可以随意选择？意义的确立在多大程度上需要社会共识？如果用文字来定义符号，是否意味着它继承了自然语言的不精确性？

This historicization of the idea of algorithm should serve as a warning against uncritically identifying the symbolic methods of the past with modern algorithms. The algorithm as we know it is a complex amalgam whose prehistory encompasses a range of practices, including astronomical and statistical computation, bureaucratic procedures, market economics, and governmental data-gathering efforts such as the US census. As a background to modern algorithms, symbolic methods are important less on account of their intrinsic relevance than because of the role they came to play in technical discourse. In the 1960s and ’70s, the discipline of computer science came to view algorithms as abstract processes that maintain a stable identity even as they are implemented, explained, applied, and interpreted in a range of ways. As I show in this book, this way of thinking is implicated in a long series of debates about the relation of symbols to language. Should the same symbols be used both to compute results and to present them to others? To what extent can their meanings be chosen at will, and to what extent does the establishment of meaning require social agreement? If a symbol is defined using words, does that entail that it inherits the imprecision of natural language?

现在，这类问题会被视为计算之外的问题，涉及人们赋予算法的意义，而不是算法本身。但这种界限并非一直存在，通过研究算法随时间的变化就可以看出这一点。印度的计算技术始终包含指令，通过直接的命令式陈述或示例来教授如何处理符号：如果总和大于 9，则在左边的数字上方写一个 1。正如人们在计算机时代之前很久就认识到的那样，这种类型的程序可能由机器执行。18然而，最初的“算法”还涉及另一种不那么明显的机械规则：9 表示九。也就是说，这种做法不仅包括如何操作符号的规则，还包括如何解释符号的规则。虽然数学家早就认识到这些语义规则不同于计算过程，但它们仍然是“算法”的一部分。

Such issues would now be seen as extrinsic to computation, involving the significance people assign to algorithms, not the algorithms themselves. But this boundary has not always been in place, as one can see by examining how what counted as an algorithm has changed over time. The Indian computational techniques have always involved instructions, taught either through direct imperative statements or by example, for what to do with symbols: if the sum is greater than 9, write a 1 above the digit to the left. As people recognized long before the computer age, this type of procedure can potentially be performed by machines.18 Yet the original “algorithm” also involved another, less obviously mechanical sort of rule: 9 means nine. The practice, that is, included rules not just for how to manipulate the symbols but also for how to interpret them. While mathematicians long recognized that these semantic rules differed from calculating procedures, they were a part of the “algorithm” just the same.

符号代数通过引入字母来表示未指定的值，例如ax + b ，从而使这些问题更加复杂。这种字母的使用由弗朗索瓦·韦达在 16 世纪 90 年代引入，它通过一种抽象的形式描述过程，使输入保持未指定，从而为现代算法奠定了基础。但这些字母与 + 和 - 等运算符连接在一起，至少在早期，这些运算符应该具有固定的含义。在简单的情况下确定这些含义可能不会造成大问题，但随着符号方法扩展到诸如微积分等理论上充满挑战的领域，事情变得更加棘手，而在莱布尼茨尝试为政治开发符号方法的乌托邦方案中，情况变得更加糟糕。例如，假设我们引入一个符号来表示公平。我们如何确保使用这个符号的每个人都同意什么是公平？概念清晰度的重要性使得人们很难忽视如何使符号具有某种意义的问题，而对这个问题的不同答案对于象征主义的作用具有强烈的暗示。

Symbolic algebra complicated these matters by introducing letters to indicate unspecified values, as in ax + b. This use of letters, introduced by François Viète in the 1590s, laid the groundwork for the modern algorithm by enabling procedures to be described in an abstract form that leaves the inputs unspecified. But these letters were linked together with operators such as + and – that were, at least early on, supposed to have fixed meanings. Establishing these meanings may not have posed a major problem in simple cases, but things became trickier as symbolic methods extended into theoretically fraught fields such as the infinitesimal calculus, and they became yet worse in utopian schemes like Leibniz’s attempt to develop symbolic methods for politics. Suppose, for instance, we introduce a symbol to denote equity. How can we be sure that everyone using this symbol agrees about what equity is? The importance given to conceptual clarity made it difficult to ignore the question of what it takes to make a symbol mean something, and disparate answers to this question had strong implications for what symbolism could do.

将意义从算法中剔除，与其说是解决了这些问题，不如说是剥夺了它们的认识论意义。这一过程的早期阶段可以在19世纪看到，当时像乔治·布尔这样的代数学家赋予了形式规则在其科学中新的基础地位。随着编程语言的发展，这一界限在20世纪得到了巩固。早期的编程语言，例如1958年首次推出的ALGOL，既提供了一种控制计算机的方法，也提供了一种发布算法的标准媒介。作为一种交流手段，编程语言通常并不独立于人类使用的语言；代码通常使用词语，包括内置关键字（如ifand）for以及用户定义的函数和变量名称，以使其工作原理更易于理解。对这些语言包含的普遍解释是，它们仅仅是为了方便理解，而不会影响算法本身，因为算法本身是根据形式语义定义的。这种“硬”算法逻辑和“软”交流问题之间的划分——这种划分渗透到了计算机科学的讨论中——使得程序员可以推进计算系统的设计，而不必担心需要如何才能建立关于意义的一致意见，如果这种一致意见确实能够建立的话。

The expulsion of meaning from algorithms did not so much resolve these issues as divest them of epistemological significance. An early phase of this process may be discerned in the nineteenth century, when algebraists like George Boole granted formal rules a newly foundational role in their science. The boundary solidified in the twentieth century with the development of programming languages. Early programming languages such as ALGOL, first introduced in 1958, provided at once a way to control computers and a standard medium for publishing algorithms. As means of communication, programming languages do not, in general, work autonomously from the languages people speak; code typically uses words, both in built-in keywords like if and for and the user-defined names of functions and variables, to make its workings easier to understand. The received explanation of these linguistic inclusions is that they are mere conveniences that aid comprehension without affecting the algorithm itself, which is defined in terms of a formal semantics. This division between “hard” algorithmic logic and “soft” communicational matters—a division that came to pervade the discourse of computer science—gives programmers license to push ahead in the design of computational systems without worrying about what it would take to establish an accord about meaning, if, indeed, this accord is ever established at all.

将符号方法与语言的关系进行历史回顾，可以发现这种思维方式并非符号方法所固有；过去的情况并非如此，未来也可能如此。像 GPT-3 这样的基于语言的人工智能系统，融合了计算逻辑和集体生成的语言数据，进一步明确了两者之间的区别。将计算和交流推向了极限，因此它们提供了一个重新思考计算过程与语言之间关系的基本假设的机会。本书的核心观点是，现代算法理念依赖于一种特殊的主客体划分：将学科的严谨标准与复杂的文化、语言和教学因素区分开来，而这些因素使系统易于人们理解。在计算机编程学科中，这些标准提供了一种思考计算过程（创建和评判计算过程）的方式，使这些过程作为数学抽象而客观存在，独立于具体的计算机系统。这种主客体划分不仅深深植根于教科书的算法定义中，也存在于现代编程语言的设计中，现代编程语言通常使算法逻辑尽可能地独立于交流问题；这种抽象促进了算法在计算机系统和应用领域之间的转移。这种思维方式直到19世纪才真正牢固确立，重新审视产生这种思维方式的条件，可以帮助我们更好地理解像GPT-3这样基于语言的机器学习系统的含义。算法的概念就像一道阻挡语言社会复杂性的堤坝，即将溃决。本书讲述的是激发它建造的那场洪水。

Historicizing the relation of symbolic methods to language shows that this way of thinking is not inherent to symbolic methods; things have been otherwise in the past, and they could be otherwise in the future. Language-based AI systems like GPT-3, with their admixture of computational logic and collectively produced linguistic data, push the distinction between computation and communication to its utmost limits, and they thus provide an occasion to reconsider fundamental assumptions about how computational processes relate to language. The central claim of this book is that the modern idea of algorithm depends on a particular sort of subject–object divide: the separation of disciplinary standards of rigor from the complex array of cultural, linguistic, and pedagogical factors that go into making systems comprehensible to people. In the discipline of computer programming, these standards provide a way of thinking about computational procedures—of creating them and judging them—that grants these procedures an objective existence as mathematical abstractions, apart from concrete computer systems. This subject–object divide is deeply embedded not just in textbook definitions of algorithm but also in the design of modern programming languages, which generally make algorithmic logic as independent as possible from matters of communication; this abstraction facilitates the transfer of algorithms across computer systems and across application domains. This way of thinking was not firmly in place until the nineteenth century, and revisiting the conditions that produced it can help us better understand the implications of language-based machine learning systems like GPT-3. The idea of algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction.

从公式到源代码

From Formulae to Source Code

本书探讨了与数学相关的语言学问题，承接了科学史学的悠久传统。20 世纪 90 年代，彼得·迪尔 (Peter Dear) 和罗伯特·马克利 (Robert Markley) 等学者开始关注语言在 17 世纪实验科学兴起中的作用。19近期学术研究则探索了过去的语言学科对数学和计算的影响。尤其值得一提的是，关于语言学与早期计算机史的交叉研究已有很多，包括语法理论对编程语言的重要性以及机器翻译作为一项研究项目的兴起。20回顾更早的时期，凯文·兰伯特 (Kevin Lambert) 和特拉维斯·D·威廉姆斯 (Travis D. Williams) 等学者探讨了数学家与语言学（长期以来一直关注数学符号的历史）以及修辞学（其技术可以在数学证明中体现）的互动。21

In broaching linguistic issues in relation to mathematics, this book joins a long tradition in the historiography of science. In the 1990s, scholars such as Peter Dear and Robert Markley drew attention to the role of language in the emergence of experimental science in the seventeenth century.19 More recent scholarship has explored the influence of linguistic disciplines from the past on mathematics and computation. There has, in particular, been a great deal of research on the intersection of linguistics with early computer history, including the importance of theories of syntax for programming languages and the emergence of machine translation as a research program.20 Looking at earlier time periods, scholars such as Kevin Lambert and Travis D. Williams have discussed mathematicians’ engagements with philology, which long concerned itself with the histories of mathematical symbols, and rhetoric, whose techniques can be discerned in mathematical proofs.21

除一些例外，数学符号主义的历史主要集中于认识论问题，例如数学证明标准的变化、符号所开启的新思维模式像a ^b这样的数字，以及数学结构如何与现实相关（或不相关）。本书探讨了这些问题，但更侧重于符号体系中相对被忽视的交流方面。正如这个词的形式所暗示的那样，交流需要人与人之间的共同点，而这种共同点在文字和符号中的作用方式是否相同，这并不是不言而喻的。几个世纪以来，人们已经认识到词语的使用在某种程度上受到惯例的制约。正如 17 世纪哲学家伯纳德·拉米所说：“如果我们愿意，我们可以把马叫做狗，把狗叫做马；但是前者的观念已经固定在马这个词上，后者的观念已经固定在狗这个词上，我们不能把它们互换，也不能把一个当作另一个，否则就会完全混淆人类的对话。” 22为了用英语有效地交流，一个人必须至少大体上遵循他人的用法。另一方面，代数符号的含义似乎取决于个人意愿：人们可以写“设a = 5”，而这正是a 的意志所指。23对许多观察家来说，这种以个人主义定义的符号似乎矛盾地传达了文字无法比拟的思想透明度。历史上的思想家以各种方式探讨了这一明显的悖论，反映了语言、知识和思维形成规则的变迁。

With some exceptions, histories of mathematical symbolism have focused primarily on epistemological matters such as changing standards of mathematical proof, the new modes of thought opened by notations like a^b, and how mathematical constructs relate (or do not relate) to reality. This book considers these matters, but it places more emphasis on the relatively neglected communicational side of symbolism. Communication, as the form of the word suggests, requires a common ground between people, and it is not self-evident that this common ground works the same way with words and symbols. For centuries, it has been recognized that the use of words is to some extent constrained by convention. As the seventeenth-century philosopher Bernard Lamy put it, “We might, if we please, call a Horse a Dog, and a Dog a Horse; but the Idea of the first being fixt already to the word Horse, and the latter to the word Dog, we cannot transpose them, nor take the one for the other, without an entire confusion to the Conversation of Mankind.”22 To communicate effectively in English, one must, at least broadly, follow the usages of others. The meanings of algebraic symbols, on the other hand, appear to bend to the individual will: one can write, “let a = 5,” and that is what a will mean.23 To many observers, such individualistically defined symbols have seemed, paradoxically, to convey ideas with a level of transparency that words could not match. Historical thinkers have addressed this apparent paradox in a range of ways, reflecting changing precepts about language, knowledge, and the formation of thought.

对这些问题的关注使得人们对代数符号在现代科学起源中的作用的思考变得复杂。长期以来，人们普遍认为，17 世纪的科学革命涉及物理科学的“数学化”。最近的趋势是认识到数学类别本身在那个时期发生了变化。2016 年编辑的《自然的语言：重新评估 17 世纪自然哲学的数学化》一书致力于更细致地理解科学数学化的含义。24本研究通过考察从近代至今人们理解数学符号的方式的变化，为这种细致入微的理解做出了贡献。虽然代数符号在 17 世纪激发了极大的热情，但我希望表明，这种热情并不总是与“数学”的概念联系在一起。早期，人们的兴奋更多地源于符号的视觉特性，它预示着一种与英语和拉丁语等口语截然不同的交流方式。因此，要理解符号方法在科学史中的地位，不仅需要对数学范畴进行历史化，还需要对语言范畴进行历史化；尤其需要思考关于书写与口语的关系以及如何建立交流的共同基础的观点的转变。

An attention to these issues complicates received thinking about the role of algebraic symbolism in the origin of modern science. It has long been a common narrative that the Scientific Revolution of the seventeenth century involved the “mathematization” of the physical sciences. The trend more recently has been toward recognizing that the category of mathematics itself changed in the period. The 2016 edited collection The Language of Nature: Reassessing the Mathematization of Natural Philosophy in the Seventeenth Century works toward a more nuanced view of what it means for a science to be mathematized.24 The present study contributes to this nuancing by examining the changing ways people made sense of mathematical symbols from the early modern period to present. While algebraic notation inspired a great deal of excitement in the seventeenth century, this excitement was not, as I hope to show, always tied to a conception of “mathematics” at all. Early on, the excitement had more to do with the visual nature of the symbols, which promised a mode of communication fundamentally different from spoken languages such as English and Latin. Understanding the place of symbolic methods in the history of science thus requires historicizing not only the category of mathematics but also the category of language; in particular, we must consider changing opinions on the relation of writing to speech and on how the common ground of communication should be established.

试图为这段历史找到一个绝对的开端是徒劳的。在我们看来，类似算法的实践至少与文字本身一样古老，并在一系列文化中独立发展。撇开一些古代和中世纪数学的背景不谈，我的叙述始于16世纪，当时现代代数符号开始被编纂。正如我在第一章中所讨论的，在16世纪中叶到17世纪中叶之间，这种符号彻底改变了代数的实践：此前，方程求解程序主要通过文字来阐述，而现在人们可以用简洁的公式来表达它们。在普遍对语言持怀疑态度的氛围中，这些符号被视为一种更优越的替代方案，一种无需文字中介，直接将思想呈现在眼前的方式。我认为，这种对符号透明性的信心源于这样一种信念：某些普遍的思想被神圣地铭刻在所有人类的脑海中，从而使得完美的交流能够不受语言偶然性的影响。

Attempting to find an absolute beginning for this history would be hopeless. Practices that look to us like algorithms have existed at least as long as writing itself, developing independently in a range of cultures. Aside from some background about ancient and medieval mathematics, my account starts in the sixteenth century, when the modern form of algebraic notation began to be codified. As I discuss in chapter 1, between the mid-sixteenth century and the mid-seventeenth, this notation revolutionized the practice of algebra: whereas equation-solving procedures had previously been expounded largely through words, one could now express them in compact formulae. Amid a general climate of suspicion toward language, such symbols came to be seen as a superior alternative, a way of presenting ideas directly to the eye without the mediation of words. This confidence in the transparency of symbols rested, I argue, on a belief that certain universal ideas were divinely etched onto all human minds, thus enabling perfect communication independently of the contingencies of language.

莱布尼茨的工作既是早期现代对符号痴迷的顶峰，也是一个转折点。在第二章中，我将讨论符号在他的数学工作和他创建推理演算器的尝试中所扮演的角色。莱布尼茨是较早将算法一词扩展为印度计算技术变体以外其他含义的作家之一（尽管不是第一个）：他使用其拉丁语和法语同源词来指代他版本的微积分中的微分过程。然而，他的意思并不完全是现代的，而对这种语义细微差别的关注揭示了算法思维史上一个经常被忽视的方面。莱布尼茨的“算法”不是以普通算术为模型，而是以符号代数为模型；它不是由确定每一步正确进行的精确定义的程序组成，而是由一组用于转换表达式的方程组成。算法的代数意义具有广泛而持久的影响，它使这一思想与新符号符号的发展密切相关。

Leibniz’s work was both a culmination of this early modern obsession with symbols and an inflection point. In chapter 2, I discuss the role of symbols in both his mathematical work and his attempt to create a calculus ratiocinator. Leibniz was one of the earlier writers (although not the first) to extend the word algorithm to something other than variants of the Indian calculating techniques: he used its Latin and French cognates to refer to the differentiation procedure of his version of calculus. Yet his meaning was not quite the modern one, and an attention to this semantic nuance reveals an aspect of the history of algorithmic thinking that has often been overlooked. Leibniz modeled his “algorithm” not on common arithmetic but on symbolic algebra; it consists not of a precisely defined procedure that determines the correct manner of proceeding at each step but rather of a collection of equations for use in transforming expressions. This algebraic sense of algorithm, which had widespread and enduring influence, placed the idea in an intimate relation to the development of new symbolic notations.

莱布尼茨在广泛的领域中尝试了这些符号，从积分的∫ dx符号到二进制数字，再到为政治和法律发展符号方法的尝试。然而，语言观念的转变正在削弱这一项目的根基。在汉娜·道森的记述中，语言思想中的关键人物是莱布尼茨的智识对手约翰·洛克。25莱布尼茨与洛克的争论通常被解读为认识论的争论，探讨的是非经验知识形式的合法性。正如我在本章中指出的那样，这场争论也对符号方法产生了影响。莱布尼茨认为，概念已经符号的意义在诞生之初就存在于心智中，因此稳定符号的意义并非难事。洛克挑战了这一假设，并由此质疑符号与文字之间是否真的存在如此大的区别。在洛克的长期影响下，数学家们更加关注概念定义；如今人们认为，清晰度并非源于符号本身，而是源于数学概念在心智中形成的方式。

Leibniz experimented with such notations in a wide range of contexts, from the ∫dx notation for integrals to binary numerals to attempts to develop symbolic methods for politics and law. Shifting ideas about language were, however, undermining the grounds of this project. In Hannah Dawson’s account, a pivotal figure in linguistic thought was Leibniz’s intellectual rival John Locke.25 Leibniz’s dispute with Locke is commonly interpreted as epistemological, dealing with the legitimacy of nonempirical forms of knowledge. The debate also, as I show in the chapter, had implications for symbolic methods. Leibniz assumed that concepts already existed in the mind at birth, so that stabilizing the meanings of symbols would not be a major problem. Locke troubled this assumption and, by doing so, called into question whether the symbols really were so different from words. In Locke’s long shadow, mathematicians paid a heightened attention to conceptual definitions; clarity, it was now believed, stemmed not from the notation itself but from the way mathematical concepts were formed in the mind.

尽管洛克语言观的兴起注定了莱布尼茨关于符号的更极端主张的失败，但它既没有扰乱符号方法的发展，也没有打乱将代数变成通用语言的愿望。在第三章中，我将重点介绍尼古拉·德·孔多塞在 18 世纪 90 年代发展起来的莱布尼茨普遍性特征的一个相对较少讨论的继承者。在法国大革命后的恐怖统治鼎盛时期，孔多塞勾勒出一个系统，为各种学科提供类似代数的符号。与莱布尼茨一样，孔多塞也致力于通过符号来解决人们的政治和文化差异。然而，他的方法却截然不同。与莱布尼茨不同，孔多塞并不假定符号表达的思想已经具有普遍性；相反，他想通过教育计划使它们具有普遍性。这种方法使他的体系过于政治化，依赖于对社会应该是什么样子的特定愿景。

Although the rise of Lockean views of language spelled doom for Leibniz’s more extreme claims about symbols, it disrupted neither the development of symbolic methods nor the desire to turn algebra into a universal language. In chapter 3, I focus on a relatively little-discussed successor to Leibniz’s universal characteristic developed in the 1790s by Nicolas de Condorcet. At the height of the Reign of Terror following the French Revolution, Condorcet sketched out a system that would provide algebra-like notations for all manner of subjects. Like Leibniz, Condorcet was out to resolve people’s political and cultural differences by means of symbols. Yet his method was very different. Unlike Leibniz, Condorcet did not presume that the ideas expressed by symbols were already universal; rather, he wanted to make them universal through a program of education. This approach rendered his system overtly politicized, dependent on a particular vision of what society should look like.

尽管孔多塞的方案几乎没有受到直接影响，但它代表了关于符号政治的争论，这种争论在计算史学中应该占有更大的地位。18 世纪数学的标准描述强调了国家传统之间的分歧：大陆数学家拥抱莱布尼茨的符号和方法，而英国数学家则追随艾萨克·牛顿拒绝它们。我在第三章中指出，18 世纪的数学受到意识形态和国家分歧的双重影响。正如索菲亚·罗森菲尔德所表明的，语言在法国大革命期间成为一个分裂的话题，因为人们将大革命的分裂归咎于未能就自由、平等和博爱等术语的含义达成一致。26这一时期语言思想的核心思想家孔狄亚克神父将代数作为解决此类分歧所需的清晰度的典范。孔狄亚克将代数视为一种“合式语言”，引发了一系列关于符号是否真的具有清晰定义的争论，例如那场关于负数是否存在的臭名昭著的争论。符号方法如何运作，在当时取决于一个关于语言政治的问题：符号的意义应该由人民集体决定，还是由学者决定。

Although Condorcet’s scheme can be assigned little direct influence, it typifies a contention over the politics of symbolism that deserves a larger place in the historiography of computation. Standard accounts of eighteenth-century mathematics emphasize a division between national traditions: Continental mathematicians embraced Leibniz’s notation and method, whereas the English followed Isaac Newton in rejecting them. I argue in chapter 3 that eighteenth-century mathematics was cut across by ideological as well as national divides. As Sophia Rosenfeld has shown, language became a divisive topic during the French Revolution, as people blamed the Revolution’s splintering on a failure to agree on the meanings of such terms as liberty, equality, and fraternity.26 A central thinker in the linguistic thought of the period, the Abbé de Condillac, held up algebra as a model of the clarity needed to resolve such disagreements. Viewing algebra as, in Condillac’s terms, a “well-formed language” led to a number of debates over whether the symbols really did have clear definitions, as in a notorious controversy over the existence of negative numbers. How symbolic methods worked hinged, in this moment, on an issue regarding the politics of language: whether the meanings of signs ought to be governed collectively by the people or decided on by the learned.

这场冲突与其说是得到了解决，不如说是被放弃了。在19世纪初世纪，代数学家将注意力从概念定义转向形式规则，这为数学严谨性提供了新的标准。在第四章​​中，我将重点介绍英国爱尔兰数学家乔治布尔的工作，他描述了最终成为布尔逻辑的系统。布尔的工作很少被认为是以莱布尼茨和孔多塞为代表的通用语言传统的一部分，通常被定位于代数和逻辑的交叉点。但在 1847 年首次介绍他的系统的书中，布尔将符号逻辑描述为“迈向哲学语言的一步”。27认真对待这一说法，我认为布尔的项目是由语言思想的另一次重大转变促成的。虽然布尔只是迷恋符号作为他的先驱，但他并不敌视文字；相反，他信奉尊重甚至崇敬人们从祖先那里继承下来的语言。这种态度使得十八世纪冲突的两个派系达成了休战协议：符号不应该取代语言，而应该与语言协同工作，同时从机械规则中汲取严谨性，从文字中汲取意义。

This conflict was less resolved than it was abandoned. In the early nineteenth century, algebraists turned their attention from conceptual definitions to formal rules, which provided a new standard of mathematical rigor. In chapter 4, I focus on the work of George Boole, the English Irish mathematician who described the system that would eventually become Boolean logic. Boole’s work has seldom been considered part of the universal language tradition exemplified by Leibniz and Condorcet, being typically positioned at the intersection of algebra and logic. But in the 1847 book in which he first introduced his system, Boole describes symbolic logic as “a step toward a philosophical language.”27 Taking this claim seriously, I contend that Boole’s project was enabled by another major shift in linguistic thought. While Boole was just an enamored with symbols as his precursors, he lacked their hostility toward words; instead, he espoused a respect and even a reverence toward the languages people inherit from their ancestors. This attitude enabled the two factions that clashed in the eighteenth century to arrive at a truce: instead of replacing language, the symbols were supposed to work together with it, at once drawing rigor from mechanical rules and meaning from words.

很快，对语言的旧敌意又在戈特洛布·弗雷格、恩斯特·施罗德和鲁道夫·卡尔纳普的著作中卷土重来，他们再次设想用符号取代文字。但即使是他们的工作也未能消除布尔时代形成的认识论分歧。在第 5 章中，我将探讨早期的编程语言 ALGOL，其名称的意思是“算法语言”——这一选择预示着“算法”一词从 1960 年左右开始被广泛采用，成为精确定义的计算过程的通用术语。ALGOL的创建者将其描述为一种“通用语言”，可以以人类可读、机器可执行的形式指定算法。28但与布尔逻辑一样，ALGOL 对通用性的主张是狭隘的。ALGOL 并没有彻底取代白话直至形成真正的人类思维，而是使用文字（通常是英语）来帮助人们理解程序。 ALGOL 中本应具有通用性的仅仅是程序的算法“本质”，这与普通语言仍然必须发挥作用的问题（例如交流和教育）有着明显的区别—​​—简而言之，与后来被称为“人为因素”的计算方面有着明显的区别。

The old antagonism toward language would soon enough return in the work of Gottlob Frege, Ernst Schröder, and Rudolf Carnap, who once again envisioned replacing words with symbols. But even their work did not undo the epistemological divisions that formed in Boole’s time. In chapter 5, I consider the early programming language ALGOL, whose name means “algorithmic language”—a choice that heralds the widespread adoption, starting around 1960, of the word algorithm as a general term for precisely defined computational procedures. ALGOL’s creators described it as a “universal language” that could specify algorithms in a form both readable by humans and executable by machines.28 But as with Boolean logic, ALGOL’s claims to universality are narrow. Rather than replacing the vernacular all the way down to the formation of actual human thought, ALGOL employs words (often in English) to help people understand programs. What was supposed to be universal in ALGOL was only the algorithmic “essence” of a program, which was distinguished sharply from issues in which ordinary language still had to play a role, such as communication and education—in short, from the aspects of computation that were coming to be known as “human factors.”

ALGOL 的例子表明，我们如今所知的算法依赖于一种划分学科界限的特定方式。当计算机科学家开始赋予“算法”一词理论意义时，他们试图识别计算系统中那些能够进行数学分析的基本要素，而不受这些机器在其社会环境中运作方式的复杂影响。这种“硬”算法问题和“软”社交问题仍然深深植根于编程语言的技术设计及其相关论述中。但这并非不可避免。在19世纪末之前，“算法”通常不被理解为排除沟通问题；在布尔时代，计算程序通常不仅包含如何处理符号的规则，还包含符号含义的规则。如何确立这种含义是一个哲学争论的问题，而关于语言的不同观点也导致了对通用计算的不同看法。

The example of ALGOL shows that the algorithm as we now know it depends on a particular way of drawing disciplinary lines. When computer scientists started giving theoretical heft to the term algorithm, they were trying to identify essential elements of computational systems that could be analyzed mathematically, in isolation from the messiness of how those machines worked in their social contexts. This division between “hard” algorithmic matters and “soft” social ones remains deeply ingrained in the technical design of programming languages and the discourse surrounding them. But it is not inevitable. Before the late nineteenth century, “algorithms” were not usually understood to exclude issues of communication; through Boole’s time, computational procedures typically included rules not just for what to do with symbols but also for what the symbols meant. How to establish this meaning was a matter for philosophical contention, and disparate views about language entailed divergent visions for what universal computation would be.

我在本书中主要强调的是这些早期的思维方式——那些与现代计算明显不同的思维方式。在科学史中，有一条方法论准则是避免落入所谓的“辉格党史”风格——也就是说，避免通过线性的进步叙事来描述历史发展，这种叙事暗中偏向于胜利的立场。数学符号史往往带有强烈的辉格主义色彩，赞扬那些使用后来成为标准的符号的作者，批评那些不使用符号的作者。我当然无意否认符号主义的优势，但我的目的与其说是赞扬它，不如说是理解它。因此，我希望描述符号的运用所带来的损失和收获。我还希望表明，符号方法并非一个固定的范畴。几个世纪以来，人们理解符号的方式经历了多次变革，而现代算法的理念是特定环境和认识论承诺的产物。

It is primarily these earlier ways of thinking—the ones that are noticeably different from modern computation—that I emphasize in this book. In the history of science, it is a methodological precept to avoid falling into the style called “Whig history”—to avoid, that is, describing historical developments through linear narratives of progress that implicitly side with the positions that won. Histories of mathematical symbols tend to be extremely Whiggish, complimenting authors who use notations that later became standard and chastising those who do not. I certainly do not mean to deny the advantages of symbolism, but my purpose is less to celebrate it than to understand it, and I accordingly hope to describe what was lost with the adoption of symbols as well as what was gained. I also hope to show that the symbolic method is not a fixed category. The ways people have understood symbols changed multiple times over the centuries, and the modern idea of algorithm is a product of particular circumstances and epistemological commitments.

另一种变化的迹象在二十一世纪初开始出现。在 2010 年代，“算法”一词越来越多地不再指 ALGOL 旨在表示的精确定义的程序，而是指像 GPT-3 这样的机器学习系统。虽然机器学习的概念自计算机时代早期就已存在，但正如我在结尾处所论证的那样，算法含义的这种转变比人们普遍认识到的更多地代表了与二十世纪概念的决裂。 29像 GPT-3这样的文本生成器有望带来一种新的编程范式，在这种范式中，程序员不是设计计算过程，而是用英语向计算机发出命令。即使对于那些完全接受这个想法的人来说（也许是少数），也很难否认它的广泛采用会重新重视语言的缺陷——即词语实际上可能不够清晰或稳定，不足以成为技术知识的适当媒介。随着机器学习的广泛应用，“硬”逻辑和“软”交流问题之间的界限变得混乱，算法成为争论的焦点。

Signs of another such change began to appear in the early twenty-first century. Over the course of the 2010s, the word algorithm came increasingly to refer not to the precisely defined procedures ALGOL was designed to represent but to machine learning systems like GPT-3. While the idea of machine learning has existed since the early computer era, this shift in the meaning of algorithm, as I argue in the coda, represents more of a break from twentieth-century conceptions than has generally been recognized.29 Text generators like GPT-3 promise a new programming paradigm in which, instead of designing a computational procedure, programmers give the computer orders in English. Even for those (perhaps a minority) who are fully comfortable with this idea, it is hard to deny that its widespread adoption would give a renewed importance to the flaws of language—to the possibility that words are not actually clear or stable enough to form an adequate medium for technical knowledge. With the widespread adoption of machine learning, the division between “hard” logic and “soft” communicational matters has become troubled, and algorithms have become a site of contestation.

这些发展虽然新颖，但在某种程度上却标志着回归到18世纪的境况，那时布尔和他的同时代人尚未在符号和语言之间设置障碍。18世纪的数学家们并不认为词语的含义与符号方法无关；相反，他们热烈地争论符号是否必须与公认的词语定义相对应，或者符号是否可以被重新定义。他们也没有将计算系统与政治割裂开来。一些人将符号视为挑战既定思维方式的一种方式，这一理念后来与法国大革命的理性化改革联系在一起。另一些人则持相反观点，他们认为词语是宝贵的遗产，其影响力对于保持数学知识与一个国家的文化相符至关重要。正如本书所旨在做的那样，关注这些早期的论述，可以让我们更好地理解计算与语言交叉领域中存在的可能性和问题。

New as these developments are, they in some ways mark a return to the situation in the eighteenth century, before Boole and his contemporaries threw up a barrier between symbols and language. Mathematicians in the eighteenth century did not view the meanings of words as irrelevant to symbolic methods; instead, they heartily debated whether symbols had to correspond to received definitions of words or whether they could be defined anew. Nor did they set computational systems apart from politics. Some viewed symbols as a way of challenging received ways of thinking, an idea that came to be associated with the rationalizing reforms of the French Revolution. Others took the opposite view, cherishing words as a precious inheritance whose influence was needed to keep mathematical knowledge in line with the culture of a country. Attending to these earlier discourses, as this book aims to do, can provide us with a better sense of the possibilities and problems that exist at the intersection of computation and language.

将这段历史看作是一系列指导术语——在特定的历史背景下，这些思想设定了评判符号方法的标准——可能会有所帮助。在十七世纪，欧洲人通常将计算描述为一种技巧或艺术，意为一套系统发展的技能。计算之所以成为一门艺术，是因为它具有可传递性：人们可以物理地演示、表达或写下正确的计算方法，从而使人们能够以可控的方式发展和练习这项技能。在十八世纪，对技巧的重视很大程度上让位于对自然理性的崇拜——这是一种重视心灵天生能力的指导原则。这种思维方式鼓励人们淡化明确的规则，而青睐概念性解释，这些解释应该使执行计算的正确方法直观地显而易见。在布尔时代，对启蒙思想的反动导致人们从自然理性转向截然相反的文化价值。在这颗恒星之下，机械必须与有机保持平衡，因此抽象的数学系统和人类思维，在社区中发展起来的语言的培育下，形成了一个整体的两半。

It may be helpful to think of this history as a succession of guiding terms—ideas that, in particular historical contexts, set the standards by which symbolic methods were judged. In the seventeenth century, Europeans typically described computation as an artifice or art, meaning a systematically developed set of skills. What made computation an art was its transmissibility: one could physically demonstrate, articulate, or write down the correct way of doing it, thus enabling people to develop and practice the skill in a controlled fashion. In the eighteenth century, the valuing of artifice largely gave way to the cult of natural reason—a guiding principle that valued the mind’s inborn faculties. This way of thinking encouraged a deemphasis of explicit rules in favor of conceptual explanations that were supposed to make the correct way of performing a computation intuitively obvious. In Boole’s time, the reaction against Enlightenment thought led to a turn away from natural reason to the quite contrary valuing of culture. Under this star, the mechanical had to be balanced with the organic, and thus abstract mathematical systems and human thought, as fostered by the languages that develop in communities, formed two halves of a whole.

虽然文化观念持续影响着计算，但指导现代算法的理念，如果真要说起来，其实是技术。“技术”一词非常古老，但它曾经的含义与现在截然不同，它既指关于某种技能实践的论文，也指讨论该实践时使用的一套技术术语。30它的现代含义在十九世纪后期占据主导地位，更多地与科学知识的实际应用有关。将计算视为技术，鼓励人们精确地定义问题，从而将系统中可以应用严格工程方法的各个方面分离出来——这是一种观点这促使早期计算机科学家将算法理论化为抽象的程序，使其能够脱离具体的应用环境进行分析。这种分而治之策略的全面影响直到21世纪初才显现出来。当时，在一个几乎抽离了所有人类经验的智力框架内发展起来的技术，成为了一股掌控世界大部分地区的力量。

While the idea of culture continues to influence computation, the idea guiding the modern algorithm is, if anything, technology. Technology is a very old word, but it once meant something very different from its present sense, referring either to a treatise about a skilled practice or to the set of technical terms used in discussing it.30 The modern meaning, which became dominant in the late nineteenth century, has more to do with the practical application of scientific knowledge. Viewing computation as technology encourages defining problems precisely so as to isolate aspects of systems that can be subjected to rigorous engineering methods—a perspective that motivated early computer scientists to theorize algorithms as abstract procedures that may be analyzed apart from the specific contexts in which they are used. The full ramifications of this divide-and-conquer strategy did not become apparent until the early twenty-first century, when techniques that were developed within an intellectual framework that abstracted out almost all human experience became a force that runs much of the world.

符号方法的历史在某种程度上与如今围绕算法的政治争论相去甚远。本书主要探讨的是通用计算理念还只是一种不切实际的臆想而非社会现实的时代。但这种臆想所引发的许多问题在计算机时代依然存在。几个世纪以来，每当符号方法探索新领域时，诸如它们是否能够或应该保持政治中立之类的问题就一次又一次地出现。然而，争论的语境千差万别，回顾早期的历史可以揭示当前话语的假设。我从近代早期开始论述，当时人们对符号方法的热情四溢——但其兴起的原因与21世纪人工智能的狂热截然不同。

The history of symbolic methods is in some ways remote from the political contentions that now surround algorithms. This book largely deals with a time when the idea of universal computation was more a matter of starry-eyed speculation than a social reality. But many of the issues that arose from this speculation have remained with us in the computer age. Questions like whether symbolic methods can or should be politically neutral have come up again and again over the centuries at moments when these methods were venturing into new territory. The terms of debate, however, have varied widely, and attending to earlier moments can be revealing about the assumptions of the present discourse. I begin in the early modern period, when excitement about symbolic methods was widespread—but for reasons quite opposed to those that have inspired the hype surrounding twenty-first-century AI.

偶像和象形文字

Idols and Hieroglyphs

在十七世纪的科学界，文字名声不佳。弗朗西斯·培根在其1623年出版的《学术的进步》一书中，对所谓的“市场偶像”——那些在日常用语中倾向于通过文字“潜入理解”的“庸俗”观念——提出了警告。1 为了抵御“名称的诱惑咒语”，他尝试性地提出了定义和“专业术语”，但即使这些还不够；他指出，要真正防止文字“对理解造成破坏”，需要“一种新的、更深刻的补救措施”。2几乎与此同时，新的数学符号也出现了爆炸式增长。3在16世纪中期，代数通常以文字的形式出现，甚至方程式（我们现在认为它们是用符号构成的）也出现在晦涩难懂的散文中。到了17世纪中期，这种冗长的文字表达方式已经被诸如ax + b = c之类的简洁符号表达式所取代。尽管培根本人对数学不感兴趣，但学者们早就注意到这些新符号与其追随者对语言的敌意之间的联系。4代数符号由托马斯·哈里奥特和勒内·笛卡尔在 17 世纪初引入，并形成了类似于现代的形式，与清晰的哲学理想联系在一起，包括 G.W. 莱布尼茨在内的众多思想家都设想为各种主题开发类似的符号。

In the scientific circles of the seventeenth century, words had a bad reputation. In the 1623 version of his book The Advancement of Learning, Francis Bacon warned against what he called the “idols of the market”—the “vulgar” notions that, in everyday speech, tend to “insinuate themselves into the understanding” by means of words.1 As a protection against “the seducing incantation of names,” he tentatively suggests definitions and “terms of art,” but even these are not enough; truly preventing words from “doing violence to the understanding,” he states, will require “a new and deeper remedy.”2 At almost exactly the same time, there was an explosion of new mathematical symbols.3 In the mid-1500s, algebra often took the form of words, with even equations, which we now think of as made out of symbols, appearing in knotty prose. By the mid-1600s, this logorrhea had given way to compact symbolic expressions like ax + b = c. Although Bacon himself had little interest in mathematics, scholars have long noted an alliance between these new symbols and his followers’ hostility toward language.4 Algebraic notation, brought into something like its modern form by Thomas Harriot and René Descartes in the early decades of the 1600s, came to be associated with a philosophical ideal of clarity, and numerous thinkers, G. W. Leibniz among them, envisioned developing analogous symbols for all manner of subjects.

本章概述了之前存在的符号方法莱布尼茨在十七世纪六十年代登场。它关注两种实践，它们最终成为现代算法思想的主要来源。第一是算法一词最初指代的一套技术。这个词（当时更常见的拼写是algorism）通常指数值计算程序，可能起源于中世纪的印度次大陆。5第二种是十七世纪初形成的代数符号体系。虽然现在把数学符号称为通用语言已经是陈词滥调，但早期现代教科书将数字和代数符号视为与字母表相当的书写形式，而不是语言。6正如语言学家阿玛莉亚·E·格纳纳德西坎 (Amalia E. Gnanadesikan) 所解释的那样，字母书写“是语言的一种转变，是一种应用于语言的技术，而不是语言本身。” 7对早期现代倡导者来说，符号有望改进书写技术，使其摆脱词语不确定性的束缚。这种观点提出了理论问题，最终在莱布尼茨和约翰·洛克之间的争论中爆发，并导致符号方法在几个世纪的哲学上引起争议。

This chapter gives an overview of the symbolic methods that existed before Leibniz’s arrival on the scene in the 1660s. It focuses on two practices that would eventually form major sources for the modern idea of algorithm. The first is the set of techniques to which the word algorithm originally referred. This word (then more commonly spelled algorism) generally referred to the procedures of numerical computation that probably originated on the Indian subcontinent in the medieval period.5 The second is the algebraic symbolism that solidified in the early 1600s. Whereas it is now a cliché to call mathematical notation a universal language, early modern textbooks presented numerals and algebraic symbols less as language than as forms of writing comparable to the alphabet.6 Alphabetical writing, as the linguist Amalia E. Gnanadesikan explains, “is a transformation of language, a technology applied to language, not language itself.”7 To their early modern advocates, symbols promised a way of improving the technology of writing so as to free it from the uncertainty of words. This view raised theoretical problems that would ultimately explode in the debate between Leibniz and John Locke, and that would render symbolic methods philosophically contentious for centuries.

符号代数的早期接受反映了数学知识概念之间的冲突。正如许多学者所指出的，关于“数学”构成的问题在当时远未得到解决；传统上，这一范畴不仅涵盖几何学和算术，还涵盖天文学和音乐，一些作家甚至将其扩展到其他实践，例如机械制造。8对当时的许多思想家来说，数学的核心是欧几里得几何。例如，伽利略·伽利莱在其1623年出版的著作《试金者》中发表了著名的“上帝用数学语言书写世界之书”的论述，他明确指的是几何图形，而非任何形式的符号符号。9在整个16和17世纪，欧洲人对代数的尊重低于几何，因为它并非传统的文科，而且被认为缺乏严格的证明标准。10符号代数改变了十七世纪的一系列实践，但其方法被广泛认为是实用的，而不是真正科学的，并且它们长期受到概念困难的困扰。

The early reception of symbolic algebra reflected a clash between conceptions of mathematical knowledge. As numerous scholars have shown, the question of what constituted “mathematics” was far from settled at the time; the category traditionally encompassed not just geometry and arithmetic but also astronomy and music, and some writers extended it to other practices such as the construction of machines.8 For many thinkers in the period, the heart of mathematics was Euclidean geometry. For instance, when Galileo Galilei made his famous statement—in his 1623 book The Assayer—that God wrote the book of the world in the language of mathematics, he was explicitly referring to geometric diagrams, not to any sort of symbolic notation.9 Throughout the sixteenth and seventeenth centuries, Europeans held algebra in lower esteem than geometry, since it was not one of the traditional liberal arts and was perceived to lack rigorous standards of proof.10 Symbolic algebra transformed a range of practices in the seventeenth century, but its methods were widely regarded as practical rather than truly scientific, and they would long be hounded by conceptual difficulties.

追溯到十九世纪 GHF Nesselmann 的著作，数学史学家用三阶段模型解释了代数符号的发展。11首先是修辞阶段，方程式完全用文字表达：“三个未知数加五等于二十”。接下来是切分阶段，一些符号被用作连字或单词的缩写：“3 co. p. 5 eq. 20”。最后，在符号阶段，符号完全取代了单词，并呈现出一种认识论作用：“3 x + 5 = 20”。该模型捕捉到了文字逐渐被符号取代的过程。一些基本的代数符号起源于缩写：罗伯特·雷科德（Robert Recorde）在其1557年的著作《维特的磨刀石》（The Whetstone of Witte）中，将“=”符号解释为“为了避免这些词语的繁琐重复：等于” 。12根据内塞尔曼（Nesselmann）的说法，这些符号最终的用途不仅仅是缩短文本，还可以通过改变页面上符号的排列来解决复杂问题。

Going back to G. H. F. Nesselmann’s work in the nineteenth century, historians of mathematics have explained the development of algebraic symbolism with a three-stage model.11 First is the rhetorical phase, in which equations are presented entirely in words: “Three unknowns plus five equals twenty.” Next is the syncopated phase, in which some symbols are used as ligatures or abbreviations of words: “3 co. p. 5 eq. 20.” Finally, in the symbolic phase, the symbols replace words altogether and take on an epistemological role: “3x + 5 = 20.” This model captures the gradualness of the process by which words gave way to symbols. Some of the basic algebraic symbols originated as abbreviations: in his 1557 book The Whetstone of Witte, Robert Recorde explains the = sign as a way “to auoid the tediouse repetition of these woordes: is equalle to.”12 Such symbols, according to Nesselmann, eventually took on uses beyond merely shortening texts, making it possible to solve complex problems by transforming arrangements of symbols on a page.

这种三阶段的描述将早期现代代数的大部分内容置于灰色地带。正如 Albrecht Heeffer 所说，Nesselmann 的年表并不清晰；切分阶段既包括像丢番图这样的古代数学家，也包括像弗朗索瓦·韦达这样的早期现代数学家，忽略了他们之间存在的数学实践的多样性。13 Nesselmann的描述还混淆了人们使用的符号（特殊符号如 + 和 = 与诸如加号和等号之类的单词）以及他们如何使用这些符号的问题。Viète 使用了一种将符号与拉丁词混合的符号，他甚至根据语法规则对其进行了变位——他使用æquatur而不是 = 。然而，他对这些半动词方程进行了基于规则的转换，就像现在在符号代数中使用的转换一样。如果我们要寻找现在称为符号操作的风格的起源，那么 Nesselmann 描述的转变并不一定是关键的。就解决问题的方法而言，使用文字、缩写或符号并没有什么区别。

This three-stage account places much of early modern algebra in a gray area. As Albrecht Heeffer has argued, Nesselmann’s chronology is unclear; the syncopated phase includes both ancient mathematicians such as Diophantus and early modern ones such as François Viète, ignoring the variety of mathematical practices that existed between them.13 Nesselmann’s account also muddles the issues of what symbols people used—special signs such as + and = versus words such as plus and equals—and how they used the symbols. Viète employed a notation that mixed symbols with Latin words, which he even inflected in accordance with the rules of grammar—instead of =, he used æquatur. Yet he subjected these semiverbal equations to rule-based transformations much like the ones now employed in symbolic algebra. If we are looking for the origins of the style now known as symbol manipulation, then the transitions Nesselmann describes are not necessarily pivotal. As far as problem-solving methods go, it makes little difference whether one manipulates words, abbreviations, or symbols.

然而，对纳塞尔曼论述的这种修正留下了一个解释上的空白。即使用符号取代文字对代数过程（即如今所谓的算法）的影响甚微，符号记号在十七世纪也并非被视为一个小小的发展。莱布尼茨远非唯一一位将符号视为通用方法基础的人；包括笛卡尔和艾萨克·牛顿在内的众多思想家都曾思考过，在其他领域发挥数字和代数符号之于数学的作用的可能性。符号记号的紧凑性是其显著优势，但仅凭这一点并不能解释人们对它的狂热程度。倡导者认为符号是一种无需中介，将思想直接写在纸上的方式；而代数记号则被广泛视为一种规避培根对市场的盲目崇拜，并赋予知识某种文字无法比拟的确定性的方式。为了理解这些态度，我们必须将符号的发展置于背景中，不仅要考虑当时的数学思想，还要考虑早期现代语言和文字的观念。

This revision of Nesselmann’s account, however, leaves an explanatory gap. Even if trading words for symbols had little effect on the procedures of algebra—on what one would now call the algorithms—symbolic notation was not viewed as a minor development in the seventeenth century. Leibniz was far from the only one to see symbols as a basis for a universal method; numerous thinkers, including Descartes and Isaac Newton, considered the possibility of doing for other fields what numerals and algebraic symbols had done for mathematics. Symbolic notation has an obvious advantage in its compactness, but this fact alone cannot explain the degree of the fervor. Advocates represented symbols as a way of putting thoughts directly on the page without mediation; algebraic notation was widely viewed as a way of circumventing Bacon’s idols of the market and granting knowledge a degree of certainty that words could not match. To understand these attitudes, we must contextualize the development of the notation not only in terms of the mathematical thought of the time, but also in terms of early modern ideas about language and writing.

要做到这一点，我们必须了解一个群体的心态，阅读和写作远不如今天这么普及。在十七世纪晚期之前，“语言”这个词主要指口头交流，正如乔纳森·霍普所说，写作对很多人来说仍然是一种“奇怪的技术”，一种有时不可靠的记录口头单词以便日后背诵的方式。14十六世纪出现了许多使写作技术更高效、更可靠的尝试。蒂莫西·布莱特在 1588 年出版的一本书中描述了一种“字符”艺术，它提供了一种“用字符进行简短、迅速和秘密的书写”的方法。15布莱特的系统包含大量的“字符”，每个字符都有一个由单词定义的“值或意义”（图 1.1）。16通过这种方式，人们只用一个字母的空间就可以写出一个完整的单词。正如詹姆斯·道格尔·弗莱明 (James Dougal Fleming) 所指出的，速记运动在近代早期完全局限于英国，但类似速记的做法在多种语言中都存在。17例如，炼金术和占星术采用了复杂的符号系统，这些符号被视为类似于密码和象形文字的秘密书写形式。18

To do so, one must step into the mindset of a population for whom reading and writing were not nearly as pervasive as they are today. Prior to the late seventeenth century, the word language primarily referred to spoken communication, and writing still seemed to many people, as Jonathan Hope puts it, “a strange technology,” a sometimes unreliable means of recording spoken words so that they could be recited later.14 The sixteenth century witnessed a number of attempts to make the technology of writing more efficient and dependable. A 1588 book by Timothie Bright describes an art of “characterie” that provides a means of “shorte, swifte, and secrete writing by Character.”15 Bright’s system consists of a large number of “Characters,” each of which has a “value, or signification” defined by a word (figure 1.1).16 By this means, one could write a whole word using no more space than a single letter. The shorthand movement was, as James Dougal Fleming has noted, entirely confined to England in the early modern period, but shorthand-like practices existed in a range of languages.17 Alchemy and astrology, for instance, employed complex systems of symbols that were viewed as secret forms of writing akin to cryptographs and hieroglyphs.18

对这些“字符”的迷恋部分源于这样一个事实，即与字母文字相比，它们将书写与口语置于不同的关系中。在近代早期，欧洲人倾向于将阅读当作一种声音来讨论，无论是字面意思还是想象中的声音。19这种思维方式有着古典的依据，尽管人们越来越认为这种依据不能令人满意。在《论解释》中，亚里士多德将书面语言的含义描述为一个多阶段的过程：字母表示（口语）单词，单词表示概念，以及（至少在某些中世纪读者的解释中）概念表示事物。20虽然最后一步在经院哲学家中争议不断，但第一步却经常被一笔带过。在近代早期的语言学思想中，通常使用字母（拉丁语为littera或litera）一词来指代字母字符和它们所代表的语音。21非语音符号似乎可以改变这种情况：例如，Recorde 的“=”并没有以任何明显的方式表示单词is equalle to的发音。亚里士多德模型并没有就此类符号提供明确的指导——等号可以理解为以 Bright 的“Characterie”的方式表示单词，或者也可以被视为绕过单词，直接切入相等的概念。

The fascination with these “characters” stemmed in part from the fact that they placed writing in a different relation to spoken language compared to alphabetical writing. In the early modern period, Europeans tended to discuss reading as if it involved a voice, be it literal or imagined.19 This way of thinking had a classical warrant, albeit one that was increasingly viewed as unsatisfactory. In On Interpretation, Aristotle describes the signification of written language as a multistage process: letters signify (spoken) words, words signify concepts, and (at least in the interpretation of some medieval readers) concepts signify things.20 Whereas the last step was endlessly controversial among the Scholastics, the first step was often glossed over. In early modern linguistic thought, it was common to use the word letter (in Latin, littera or litera) to refer indifferently to both alphabetical characters and the speech sounds they represent.21 Nonphonetic symbols would seem to change this situation: Recorde’s “=,” for instance, does not in any obvious way represent the sounds of the words is equalle to. The Aristotelian model provides no clear guidance as to such symbols—the equals sign could be taken to signify the words in the manner of Bright’s “Characterie,” or else it could be seen as bypassing words and cutting straight to the concept of equality.

在数字和代数符号方面，后者占了主要优势。与字母文字不同，这些符号可以用多种语言大声朗读：英语使用者将“9 - 1”读作“九减一”，而法语使用者则将其读作“neuf moins un”，其含义在两种情况下似乎都是一样的。早期现代思想家借用经院哲学的术语，通过区分名词字符（代表词语的声音）和实字符（直接指代思想或事物）来解释这些跨语言符号。实字符的概念最著名的例子是培根1623年的《文言文的进展》，其中特别提到了汉字——汉语字符的一个子集，可以用日语或英语阅读。汉语。22培根将此特性解释为与亚里士多德模型的不同：汉字“表达的不是字母（即语音）或词语，而是事物和概念；因此，许多国家，尽管语言迥异，但在使用这些汉字方面却达成了共识，通过书写进行交流。” 23因此，“用这些汉字写成的书，每个国家都可以用各自的语言阅读和解读。” 24

In the case of numerals and algebraic symbols, there was a major mark in favor of the latter. Unlike alphabetical writing, these symbols could be read aloud in multiple languages: English speakers read “9 – 1” as nine minus one, whereas French speakers read it as neuf moins un, and the meaning appears to be the same in both cases. Borrowing from the Scholastic terminology, early modern thinkers explained such translinguistic symbols by distinguishing between nominal characters, which represented the sounds of words, and real characters, which referred directly to ideas or things. The idea of a real character appears most famously in Bacon’s 1623 Advancement of Learning, where it had particular reference to kanji—a subset of the Chinese han characters that can be read in either the Japanese or the Chinese language.22 Bacon explains this quality as a departure from the Aristotelian model: the characters “express, not their letters [i.e., speech sounds] or words, but things and notions; insomuch, that numerous nations, though of quite different languages, yet, agreeing in the use of these characters, hold correspondence by writing.”23 As a result, “a book written in such characters may be read and interpreted by each nation in its own respective language.”24

虽然培根在文章中没有讨论数学符号，但后来人们习惯于引用数字和代数符号作为真实字符的例子。25例如，数学家约翰沃利斯写道，像汉字一样，代数符号“很少需要文字的介入来表达它们的含义，以至于当不同的人用文字表达这些字符的含义时，他们很少会对相同的文字达成一致，尽管他们都表达了相同的含义，就像两个翻译家将同一本书翻译成另一种语言一样。” 26罗伯特胡克对“算术数字”做了类似的比较。27由于它与现代语言学对文字的理解不符，许多现代学者认为真实字符的想法是错误的或荒谬的；Jaap Maat 甚至称这个想法为“神话”。28但当这个想法应用于数学符号时，它就不是完全没有意义的。将这些符号称为真正的字符就等于声称它们表达了所有人都能理解的普遍思想，无论人们在发音时选择什么词——英语单词minus和法语单词moins实际上有着共同的含义，通过这个含义，减号可以用来在不同语言中清楚地传达减法的概念。29

Although Bacon does not discuss mathematical symbols in the passage, it later became routine to cite numerals and algebraic symbols as examples of real characters.25 For instance, the mathematician John Wallis wrote that, like Chinese characters, algebraic signs “so little need the intervention of Words to make known their meaning, that, when different persons come to express, in Words, the sense of those Characters, they will as little agree upon the same Words, though all express the same sense, as two Translators of one and the same Book into another Language.”26 Robert Hooke made a similar comparison with regard to “Arithmetical Figures.”27 Since it does not align with how modern linguistics understands writing, a number of modern scholars have dismissed the idea of a real character as mistaken or absurd; Jaap Maat goes so far as to call the idea a “myth.”28 But the idea is not wholly senseless when applied to mathematical symbols. Calling these symbols real characters amounts to claiming that they express universal ideas that are accessible to all people, regardless of what words one chooses when pronouncing them—that the English word minus and the French word moins really share a common core of meaning by means of which the minus sign can be used to communicate the idea of subtraction clearly across languages.29

这并不是说“真实字符”的理念仍然站得住脚。这一理念源于一种信仰，即人类心智是神圣的，用来反映世界，这排除了对文化多样性的任何严肃认识。在17世纪，并非所有人都这样认为。但从某种意义上说，数字和代数符号实际上更像是字母表的替代品，而非一种语言。正如下一节所示，17世纪的教科书教授数字的方式强调实际的书写技能和大声朗读数字。这种教学法与学习阅读和写作（或学习像布莱特所说的“字符”那样的速记）的共同点，而非学习第二语言。性别方面也存在差异：白话语言主要从女性——母亲和保姆——那里学习，而写作和数学通常都是从男性教师那里学习的。30研究数字是如何被教授的，有助于我们理解为什么像莱布尼茨这样的早期现代哲学家如此重视符号的力量，以及他们那个时代的算法与我们现在所知的算法思维不同。

This is not to say that the real-character idea still holds water. The idea depended on a faith that the human mind was divinely constructed to mirror the world, which precludes any serious recognition of cultural diversity. Not everyone thought this way in the seventeenth century. But numerals and algebraic symbols really are, in a sense, more like an alternative to the alphabet than a language. As the next section shows, seventeenth-century textbooks taught numerals in a way that emphasized physical pen skills and speaking numbers aloud. This pedagogy had more in common with learning to read and write (or with learning a shorthand like Bright’s “characterie”) than with learning a second language. There was also a difference in regard to gender: vernacular tongues were largely learned from women—from mothers and nurses—whereas writing and mathematics alike were both typically learned from male teachers.30 Examining how numerals were taught sheds light on why early modern philosophers such as Leibniz put so much stock in the power of symbols—and on how the algorism of their time was different from algorithmic thinking as we now know it.

算法的意义

The Meaning of Algorism

计算与书写之间并非必然联系。前现代文化发展出了各种各样的计算工具，从算盘到印加人开发的复杂的多层计数表。然而，印地语 - 阿拉伯数字系统的兴起使计算与书写紧密相关，并使其长期受到语言哲学争论的影响。该系统可能起源于印度次大陆；天文学家婆罗摩笈多在公元 628 年左右用梵语诗句描述了它，尽管有证据表明它在他的著作之前就已经在使用。31这些数字后来传到了阿拉伯世界，由数学家和天文学家穆罕默德·伊本·穆萨·花拉子密描述了它们。32虽然花拉子密并没有发明这些方法，但他的名字启发了算法一词，因此值得考虑他的著作的内容。

That computation has anything to do with writing is not a given. Premodern cultures developed a wide array of calculating implements, from abaci to the intricate, multilevel counting tables developed by the Inca. The rise of the Hindi–Arabic numeral system, however, gave computation a strong link to writing that would long implicate it in philosophical debates about language. This system probably originated on the Indian subcontinent; the astronomer Brahmagupta described it in Sanskrit verse around 628 CE, although there is evidence that it was already in use prior to his work.31 The numerals later spread to the Arabic world, where they were described by the mathematician and astronomer Muḥammad ibn Mūsā al-Khwārizmī.32 While al-Khwārizmī did not invent these methods, it was his name that inspired the word algorithm, and so it is worth considering the contents of his work.

花拉子密可能出生于现在的乌兹别克斯坦，他在巴格达的智慧宫图书馆任职，并在那里撰写了大量文章。33 他大约在820年完成的《完成与平衡计算概要》主要讲述解方程式；“代数”一词源于该书阿拉伯语标题中的al-jabr （通常翻译为“完成”） 。34不幸的是，他关于算术的著作只保存在不可靠的拉丁语译本中。35这些译本中最著名的一个有时被称为“Dixit Algorizmi”（“阿尔戈里兹米说”），因为译者在前两段的开头插入了这个短语；这个抄写员的偶然发现被认为是拉丁语单词algorithmus的起源，因此最终也是英语单词algorithm的起源。

Probably born in what is now Uzbekistan, al-Khwārizmī secured a position at the House of Wisdom, a library in Baghdad, where he wrote on a number of topics.33 His c. 820 Compendious Book on Calculation by Completion and Balancing is largely about solving equations; the word algebra is derived from the word al-jabr (usually translated as “completion”) in the Arabic title of this book.34 His work on arithmetic, unfortunately, survives only in unreliable Latin translations.35 The most famous of these translations is sometimes called “Dixit Algorizmi” (“Algorizmi said”) because the translator inserted that phrase at the beginnings of the first two paragraphs; this bit of scribal happenstance is thought to be the origin of the Latin word algorithmus and thus, ultimately, of the English word algorithm.

根据现存译本的资料，花拉子密的这本算术书介绍了加法、减法、乘法、除法、减半和加倍的运算方法。36在《算术集》版本中，对加法的解释是这样开始的：

From what we can gather from the surviving translations, al-Khwārizmī’s arithmetic book presented procedures for addition, subtraction, multiplication, division, halving, and doubling.36 In the “Dixit Algorizmi” version, the explanation of addition begins like this:

你将把每个位加到它所属类型的上一个位上，即个位加到个位，十位加到十位。当十位中有一个位，即个位、十位或其他位，已经加到十位时，就用一个一代替它，并将其提升到更高的位上。也就是说，如果第一个位，也就是个位，已经有十了，就把它变成一个一，并将其​​提升到十位，这样它就表示十了。但如果剩余的数字小于X，或者本身小于X，则将其保留在原处。如果没有剩余，则画一个圆圈（即o），使该位置不空。37

You will add each place to the place that is above it with regard to its own kind, i.e., units to units and tens to tens. When ten has been collected in one of the places, i.e., in the place of the units or tens or in some other place, put a one instead of it and elevate it to a higher place, i.e., if you have ten in the first place which is the place of the units, make a one of it and raise it to the place of the tens and there it will signify ten. But if there remains something from the number that is less than X or the number itself is less than X, leave it in the same place. And if nothing remains, put a circle (i.e., o), so that the place may not be empty.37

这一程序包含了许多最终以花剌子米的名字命名的思想风格的特征，包括使用条件、如果-那么逻辑，甚至我们可能认为是一个循环：“也这样做，”我们被指示，“在所有地方也这样做。” 38这是一种原型算法：早期的计算机科学家将其作为机器可以通过编程执行的程序类型的模型。

This procedure contains many of the hallmarks of the intellectual style that eventually came to bear al-Khwārizmī’s name, including the use of conditional, if–then logic and even what we might think of as a loop: “Do likewise,” we are instructed, “also in all the places.”38 This is an archetypal algorithm: early computer scientists took it as a model for the sort of procedure that a machine can be programmed to perform.

在整个十七世纪， “算法”（algorism ）或“算法”一词仍然主要指这类特定程序的变体。（虽然带有th的拼写早在15世纪80年代就出现在拉丁语中，并在16世纪50年代出现在英语中，但为了避免混淆，我将把这些历史实践统称为“算法” 。39）一些作者将“算法”与“算术”（arithmetic）区分开来，后者是中世纪哲学家波爱修斯（Boethius ）提出的七门文科之一。波爱修斯意义上的算术与各种类型的数字有关：平方数、素数、完全数，以及一系列不太为人所知的其他数字。40正如伊丽莎白时代的博学者约翰·迪所说，这项研究的目的是“在精神上崛起、攀登、上升和攀登（带着思辨的翅膀），在创世之镜中凝视形式的形式，所有可数事物的典范数量：看得见的和看不见的，凡人的和不朽的，物质的和精神的。” 41与这些崇高的学说相反，算法被视为数学的较低分支之一。虽然有时在拉丁学校教授计算艺术，但计算艺术主要是 EGR Taylor 称为“数学从业者”的（大多数）人的事——他们通过教科书、讲课和辅导独立于大学系统教授数学。42他们教授的方法可应用于航海、贸易和金融，砌砖和建筑等手工艺行业，以及弹道学等军事实践。

Through the seventeenth century, the word algorism or algorithm still referred primarily to variants of this particular set of procedures. (While the spelling with a th appeared in Latin as early as the 1480s and in English in the 1650s, I will refer to these historical practices as algorism so as to avoid confusion.39) Some authors distinguished algorism from arithmetic, which was one of the seven liberal arts set out by the medieval philosopher Boethius. Arithmetic, in the Boethian sense, was about types of numbers: squares, primes, perfect numbers, and a range of others that are less well remembered.40 As the Elizabethan polymath John Dee put it, the purpose of this study was “arise, clime, ascend, and mount vp (with Speculatiue winges) in spirit, to behold in the Glas of Creation, the Forme of Formes, the Exemplar Number of all thinges Numerable: both visible and inuisible, mortall and immortall, Corporeall and Spirituall.”41 In contrast to such lofty doctrines, algorism was seen as one of the lower branches of mathematics. While it was sometimes taught in Latin schools, the art of computation was primarily the business of the (mostly) men E. G. R. Taylor dubbed “mathematical practitioners”—people who taught mathematics independently of the university system through textbooks, lecturing, and tutoring.42 The methods they taught had applications in navigation, trade, and finance, in artisanal trades such as bricklaying and construction, and in military practices such as ballistics.

在这些实践领域，印地语-阿拉伯语算法与一系列其他形式的计算相互竞争，有时也相互协作。从业者出售诸如扇形尺之类的工具，它由一对铰接的刻有刻度的尺子组成，可用于进行近似计算。算盘和计数石历史悠久，至今仍有人钟爱它们；计数石在德国尤为流行，而算盘在东欧则持续使用了几个世纪。计算还可以涉及数表，人们可以在其中查找某些值，而无需自己计算。17世纪初，随着对数表的发展（于1614年引入），这类技术取得了重大进展。作者：约翰·纳皮尔。43由于两个数的对数相加可以得出它们乘积的对数，因此可以使用对数表将乘法简化为更简单的加法运算。这项技术是当时一些最先进的数学工具的基础，例如威廉·奥特雷德的“比例圆”，它是他于1632年描述的计算尺的前身。44

In these practical fields, the Hindi–Arabic algorism competed with and sometimes worked together with a range of other forms of computation. The practitioners sold instruments such as the sector, which consisted of a hinged pair of rulers inscribed with scales that could be used to perform approximate calculations. Abaci and counting stones had a long history, and some people continued to prefer them; counting stones were especially popular in Germany, and abaci would remain in use for centuries in eastern Europe. Calculation could also involve numerical tables, in which one could look up certain values without having to compute them oneself. The early seventeenth century saw a major advance in such techniques with the development of logarithm tables, which were introduced in 1614 by John Napier.43 Since adding the logarithms of two numbers produces the logarithm of their product, one could use logarithm tables to reduce multiplication to the much easier operation of addition. This technique was the basis of some of the period’s most advanced mathematical instruments, such as William Oughtred’s “circles of proportion,” a precursor of the slide rule that he described in 1632.44

除了教授和销售仪器外，从业者还出版书籍，至少在原则上，人们可以通过这些书籍学习计算的艺术。这些书籍大多以“普通算法”开篇，即普通计数数字的使用。该学说始于“计数”或“符号”，即学习数字的含义；之后讨论加、减、乘、除等运算。45 具体的内容各不相同。一些文本沿袭花拉子密的思想，加入了减半和加倍的特殊程序，而另一些文本则添加了开方等进一步的运算。一些教科书还包含用于货币计算的特殊“算法”，以及用于有理数和小数的更高级的算法。算法还包括旨在验证结果的附加程序，因为（毫无疑问）人类计算机经常会犯错误。

Apart from teaching and selling instruments, the practitioners also published books from which one could, at least in principle, learn the art of calculation. Most of these books started with “common algorism,” meaning the use of ordinary counting numbers. This doctrine began with “numeration” or “notation,” which meant learning the meanings of the digits; afterward came discussions of operations such as addition, subtraction, multiplication, and division.45 The exact list varied. Some texts followed al-Khwārizmī in including special procedures for halving and doubling, and some added further operations such as extracting roots. Some textbooks also included special “algorisms” for calculations involving currency as well as more advanced ones for rational numbers and decimal fractions. Algorism also included additional procedures intended to verify results, since (no doubt) human computers would often make mistakes.

这些书中描述的程序类似于现代算法，它们使用严格定义的步骤，这些步骤以符号排列开始和结束。正是这种刚性使程序具有机械性，启发了布莱斯·帕斯卡和莱布尼茨制造计算机器。但算法的第一部分 - 计数 - 是不同的。根据莱布尼茨在学校遇到的约翰·兰兹 1616 年的算术教材，计数是“对所提出的任何数字的发音和表达”。46 （拉丁语enunciatio可以表示发音或命题，暗示着与希腊语logos类似的词语和思想的混合。）第一步是熟悉九个数字 1、2、3、4、5、6、7、8 和 9 以及“密码”0，它被视为一个没有固有含义的占位符。学生必须能够识别和可靠地写下这些符号；因此，正如杰西卡·奥蒂斯（ Jessica Otis）所说，算法需要基本的书写技能，而这些技能是读写能力的一部分。47他们还必须学习符号的值，这些值通常是通过类似于布莱特（Bright）定义其速记字符“含义”的表格来教授的（图1.2）。他们还必须学习数字组成的规则，以便将它们翻译成语言中的数字词，例如84变成“八十四”。

The procedures described in these books resemble modern algorithms in their use of rigidly defined steps that begin and end with arrangements of symbols. It is this rigidity that gives the procedures the mechanical quality that inspired Blaise Pascal and Leibniz to build calculating machines. But the first part of algorism—numeration—is different. According to Johann Lantz’s 1616 arithmetic text, which Leibniz encountered in school, numeration is “the enunciation and expression of whatever number is set forth.”46 (The Latin enunciatio can mean either pronunciation or proposition, suggesting a conflation of words and ideas similar to that of the Greek logos.) The first step was to familiarize oneself with the nine digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 as well as the “cipher” 0, which was viewed as a mere placeholder that had no inherent meaning. Students had to be able to recognize and inscribe these symbols dependably; algorism thus, as Jessica Otis has argued, required the basic pen skills that were a part of literacy.47 They also had to learn the symbols’ values, which were often taught through tables similar to the one by which Bright defined the “significations” of his shorthand characters (figure 1.2). They also had to learn the rules by which numerals are composed into numbers so that they could translate them into the number words of a language, as 84 becomes “eighty-four.”

莱布尼茨详细研究的另一部著作中，对这一译本进行了更详细的讨论：约翰·海因里希·阿尔斯泰德 1630 年编著的《百科全书》。48阿尔斯泰德在赫尔伯恩学院接受教育，该学院是泛智论的中心。这是一场强调让所有人都能获得知识的教育运动；他的百科全书为莱布尼茨自己的百科全书事业树立了典范。莱布尼茨认为阿尔斯泰德对数学的处理仅仅是“他那个时代的平均水平”，但它作为十七世纪中期典型数学处理方式的一个例子还是有用的。49在其关于算术的章节中，阿尔斯泰德指出，这些数字就像一个“算术字母表”。50像兰兹一样，他将理解这个字母表与将符号翻译成文字联系起来：他写道，计算就是“正确发音书写正确的数字”。51他解释了这种翻译的几种技巧，包括标记符号以便于翻译的方法：例如，为了帮助理解 89765878910，可以在第一个数字之后的每隔三个数字上方或下方画点，从右到左，就像。

An extended discussion of this translation appears in another text Leibniz studied in detail: Johann Heinrich Alsted’s 1630 Encyclopaedia.48 Alsted was educated at the Herborn Academy, which was a center of pansophism, an educational movement that emphasized making knowledge accessible to all; his encyclopedia provided a model for Leibniz’s own encyclopedic endeavors. Leibniz judged Alsted’s treatment of mathematics to be merely “average for his time,” but it is useful as an example of what would have been considered typical in the mid-seventeenth century.49 In his chapter on arithmetic, Alsted states that the digits are like an “Arithmetical alphabet.”50 Like Lantz, he associates understanding this alphabet with translating the symbols into words: numeration, he writes, is “the right enunciation of rightly written numbers.”51 He explains several techniques for this translation, including ways of marking the symbols up so as to make the translation easier: to help make sense of 89765878910, for instance, one can draw dots above or below every third digit after the first, going right to left, as in .

虽然阿尔斯泰德关注数字的“正确发音”，但他的观点并非只有一种正确的表达方式。他观察到，希腊人和罗马人用各种方式表达数字，这些方式通常比现代拉丁语的计数法冗长得多。根据阿尔斯泰德的说法，老普林尼可能将薛西斯军队的士兵人数5,283,220表达为（大致翻译）“五十乘以二十八万三千二百二十”，而现在人们会写成“五千二百八十三千二百二十”。52同样，标记程序也会根据操作方式产生不同的读法：例如，表示为“一万千千”，而表示为“一百十万千千”。53阿尔斯泰德告诉我们，将符号按三分组是最简单的方法，但他明确表示，无论选择哪种词语，数字本身都保持不变。

While Alsted is concerned with the “right enunciation” of numbers, his point is not that there is only one correct way to do it. The Greeks and Romans, he observes, expressed numbers in various ways that are often much more verbose than the numeration of modern Latin. According to Alsted, Pliny the Elder might have expressed the number of soldiers in Xerxes’s army, 5,283,220, as (to translate loosely) “fifty times and twice a hundred and eighty-three thousands, two hundred twenty,” whereas one would now write “five double thousands, two hundred eighty-three thousands, two hundred twenty.”52 Likewise, markup procedures can produce different readings depending on how they are done: leads to “ten thousand thousand thousand,” whereas leads to “a hundred hundred thousand thousand.”53 Grouping the symbols by threes is, Alsted tells us, the easiest way of doing it, but he makes it clear that, whichever words one chooses, the number itself remains the same.

在描述语言与数学在近代早期的关系时，符号与文字之间的这种中介作用被忽视了。沃尔特·翁 (Walter Ong) 和罗伯特·马克利 (Robert Markley) 都认为，近代数学是反对话的——它抑制了古典思想所重视的来回交流。54在《斐德罗篇》中，柏拉图让苏格拉底论证说，口语优于书写，因为在对话中，人们可以动态地回答问题。55当人们从纸上读出解释时，词语总是一样的，但当人们解释自己真正理解的东西时，每次说出来的话都会不一样。用对话的能力来衡量智力（这个想法一直延续到艾伦·图灵），意味着仅仅手头有书面文本并不能保证人们知道任何事情。人们可能会认为，数字与语音的疏远口语的表达方式，甚至比字母书写更违背苏格拉底式的知识观念。但在算法文本所唤起的教学场景中，情况恰恰相反。与翁将数学与“沉默的物体世界”联系起来的观点相反，数字不应该被默默地思考。56相反，它们使老师能够鼓励学生说话：当然，是那么多千，但有多少个百呢？有多少个十呢？

This mediation between symbols and words has been overlooked in accounts of how language and mathematics related in the early modern period. Walter Ong and Robert Markley each have argued that early modern mathematics was antidialogic—that it suppressed the back-and-forth interchange that was valued in classical thought.54 In Phaedrus, Plato has Socrates argue that speech is superior to writing because, in conversation, one can dynamically respond to questioning.55 When one reads an explanation off a sheet of paper, the words are always the same, but when one explains something one genuinely understands, the words come out differently every time. Measuring intelligence by the ability to hold a conversation (an idea that persisted all the way to Alan Turing) implies that merely having a written text at hand is no guarantee that one knows anything. One might suppose that numerals, in their alienation from the phonetics of spoken languages, are even more inimical to this Socratic conception of knowledge than alphabetical writing. But in the pedagogical scene evoked by algorism texts, the opposite is the case. Contrary to Ong’s association of mathematics with a “silent object world,” numerals were not supposed to be contemplated mutely.56 Instead, they enable the teacher to prod the student to speak: sure, it is that many thousands, but how many hundreds? How many tens?

然而，这种动词化的泛滥也有其局限性。把4206读作“四千二百零六”或“四十二零六”或许同样可以接受，但读作“十二”则完全错误。教会人们理解数字需要就其价值观达成一致，这为对话的进行提供了共同的基础。阿尔斯特德在这段文字中并没有直接引用培根关于真实字符的概念，但他对数字“意义”的阐述揭示了一种类似的思维方式。就像汉字一样，数字使来自不同国家的人们能够“阅读和解读”（培根所说的）他们自己语言的文本；然而，无论人们选择哪种语言或哪种特定的措辞，这些符号所指代的数字都保持不变（我们应该这样假设）。因此，数字将通常由拉丁语等语言处理的符号功能转移到了书面交流的基本操作系统上：字母表本身。

Yet this proliferation of verbalizations had its limits. It may be equally acceptable to read 4206 as “four thousand two hundred and six” or as “forty-two oh-six,” but reading it as “twelve” would simply be wrong. Teaching people to understand digits required establishing an accord about their values, which provided the common ground on which the dialogue took place. Alsted does not directly reference Bacon’s idea of a real character in this passage, but his account of the “signification” of numerals reveals a similar way of thinking. Just like kanji characters, numerals enable people from many nations to “read and interpret,” as Bacon put it, a text in their own languages; yet the numbers to which the symbols refer remain the same (we are supposed to assume) regardless of what language or what specific wording one chooses. The numerals thus offload, as it were, the signifying function that is ordinarily handled by languages like Latin onto the basic operating system of written communication: the alphabet itself.

这一背景使我们更容易理解莱布尼茨为何坚信“计算”的力量可以扩展到其他领域。虽然推理演算（calculus reciprocator）经常与莱布尼茨在计算机器方面的工作相提并论，但两者截然不同，而且他希望用来解决争议的“计算”是否仅仅涵盖机器能够执行的计算方面尚不明确。知道如何正确地进行计算——能够在计算开始时选择正确的数字，并在计算结束时正确读出结果——也是他那个时代计算的意义之一，而使用机器并不意味着这些知识变得无关紧要。至关重要的是，计算比运算规则更少机械性，但这并不意味着它的确定性更低。阿尔斯特德对计算的论述表明，印地语-阿拉伯语系统能够以一种毫无疑问的方式表达数字。因此，这让我们有理由认为符号可以将类似的确定性扩展到其他领域。

This background makes it easier to understand Leibniz’s confidence that the power of “calculation” could be extended to other areas. While the calculus ratiocinator is often discussed together with Leibniz’s work on calculating machines, the two are distinct, and it is not clear that the “calculation” by which he hoped to settle disputes was supposed to encompass only those aspects of calculation that a machine could perform. Knowing how to numerate correctly—being able to choose the right digits at the beginning of the process and read off the results correctly at the end—was also part of what calculation meant in his time, and using a machine did not render this knowledge irrelevant. Crucially, the fact that numeration was less mechanical than the rules of operation did not imply that it was any less certain. Alsted’s account of numeration suggests that the Hindi–Arabic system can express numbers in such a way that there can be no doubt about their true values. It thus provides reason to think that symbols could extend a similar level of certainty to other areas.

然而，这种对符号力量的信心，取决于这些符号所指代的思想或事物的稳定性。人们可能会用英语的eight或拉丁语的octo来发音符号8 ，但这两个词的含义真的相同吗？如果我们把这些语言放在它们的历史背景中，它们可能并非如此：罗马人不像我们今天这样理解数字。这似乎是一个迂腐的区别，但在比普通算术更具有争议的领域，问题更加明显。例如，如果我们设计一个字符来表示恩典，那么远不是每个人都会以相同的方式理解它的含义。如下一节所示，这些问题并不特定于莱布尼茨这样的乌托邦方案。它们也出现在花剌子密数学著作的另一个主要部分：代数。虽然代数有明确的实际用途，但它提供的答案无法像算法的结果那样“说出”。因此，它的采用扰乱了数字在符号和语言之间建立的和平关系，引发了理论难题，这些难题在几个世纪以来的符号方法讨论中反复出现。

This confidence in the power of symbols depends, however, on the stability of the ideas or things those symbols are supposed to signify. One might pronounce the symbol 8 with either the English eight or the Latin octo, but are the meanings of these two words really the same? If we place the languages into their historical contexts, they are arguably not: the Romans did not understand numbers the same way we do today. This may seem a pedantic distinction, but the problem is more glaring in domains that are more obviously contentious than common arithmetic. If we devise a character, for instance, to signify grace, it is far from a given that everyone will understand its meaning in the same way. As the next section shows, such issues were not specific to utopian schemes like Leibniz’s. They also arose in another major part of al-Khwārizmī’s mathematical work: algebra. While it had clear practical use, algebra also presented answers that could not be “said” in the way the results of algorism could. Its adoption thus unsettled the peaceful relationship numerals instated between symbols and language, raising theoretical difficulties that would recur in discussions of symbolic methods for centuries.

难以言喻的数字

Unspeakable Numbers

代数（algebra）和算法（algorithm）这两个词源于同一个人，但这并不意味着它们之间有任何必然联系。如果代数（algebra）指的是一种解方程的方法——在19世纪之前，这个词几乎一直都是这个意思——那么它的历史可以追溯到几千年前。几何方程求解方法在美索不达米亚和古希腊就已经存在；中国数学家大约在公元前200年开始发展用于方程求解的数值技术。57亚历山大的丢番图在公元三世纪预见了一些符号代数的元素。58这类实践是算法性的，这远非定论，而现代代数确实包含了许多非算法性的内容。但代数在算法思维的发展中确实发挥了重要作用。花拉子密在阐述代数时，首先将其置于计算的范畴：“当我思考人们在计算中通常想要什么时，我发现它总是一个数字。” 59代数，以这种形式出现，是一种产生数字的计算艺术——它并不等同于算法，而是可以与算法协同工作的艺术。

The fact that the words algebra and algorithm derive from the work of the same person should not be taken to mean they have any necessary connection. If algebra means a way of solving equations—which is what the word almost always meant before the nineteenth century—then its history goes back millennia. Geometric equation-solving methods existed in Mesopotamia and ancient Greece; Chinese mathematicians began developing numerical techniques for equation solving around 200 BCE.57 Diophantus of Alexandria anticipated some elements of symbolic algebra in the third century CE.58 That such practices are algorithmic is far from a given, and modern algebra, indeed, contains much that is not. But algebra did play an important role in the development of algorithmic thinking. Al-Khwārizmī opens his account of algebra by placing it under the scope of calculation: “When I considered what people generally want in calculating, I found that it always is a number.”59 Algebra, in this form, is an art of calculation that produces numbers—not one that is identical to algorism but one that can work together with it.

花拉子密版本的代数在《完成与平衡计算简明书》中得以保留。该书开篇给出了方程求解的定义和一般技巧，并给出了几何证明，这些证明是按照美索不达米亚几何风格进行的。60两种基本技巧是al-jabr ，意思是完成或恢复，以及al-muqābala ，在本文中被翻译为平衡。虽然“代数”一词源于al-jabr，但现在它包含这两种技巧——将项从等式的一边移到另一边（al-jabr）和取消项（al-muqābala）。利用这两种技巧，花拉子密描述了解六种方程式；然后，他讨论了商业计算、几何图形面积计算技巧以及涉及继承的计算。最后一部分表明了代数在花剌子密的语境中的用途。阿拔斯王朝的继承规则很复杂，需要满足某些类似方程式的条件；花剌子密对这些问题的讨论几乎占据了本书的一半篇幅。61

Al-Khwārizmī’s version of algebra survives in the work titled The Compendious Book on Calculation by Completion and Balancing. His book begins with definitions and general techniques for equation solving along with geometric proofs, which he performed in the style of Mesopotamian geometry.60 The two basic techniques are al-jabr which means completion or restoration, and al-muqābala which is translated in this context as balancing. While the word algebra derives from al-jabr, it now encompasses both of these techniques—moving terms from one side of an equation to the other (al-jabr) and canceling terms (al-muqābala). Employing these two techniques, al-Khwārizmī describes procedures for solving six types of equation; he then discusses mercantile calculations, techniques for computing the areas of geometric figures, and calculations involving inheritance. This final section is indicative of the uses to which algebra was put in al-Khwārizmī’s context. The Abbasid Caliphate had complex rules for inheritance that required fulfilling certain equation-like conditions; al-Khwārizmī’s discussion of these issues takes up almost half the book.61

与现存的花剌子密算术译本一样，这本代数著作旨在向读者传授一套实用方法。与算法的直接指导不同，他通过具体问题解释他的代数方法，例如“一个平方的一半和五个根等于二十八迪拉姆（一种货币单位）。” 62用现代符号表示就是x ² / 2 + 5 x = 28。他对这个问题的解答如下：“你的首要任务是完成你的平方，使它等于一个完整的平方。你可以通过将其加倍来实现。因此，将它加倍，并将加上的数以及等于它的数也加倍。这样你就得到了一个平方和十个根，等于五十六迪拉姆。” 63他在这里指示我们做的——将系数加倍——是针对这个问题的：如果第一个数字不是一半，就必须乘以其他数字。然而，这个例子是为了代表更广泛的一类问题。他总结道：“每当你遇到等于简单数字的平方和根时，就这样做：因为它总能回答。” 64花拉子密对这些指示的编纂标志着维克多·J·卡茨和卡伦·亨格·帕沙尔所说的代数史上“算法阶段”的开始——在这个阶段，代数主要由解决特定类型问题的程序组成。65

Like the surviving translations of al-Khwārizmī’s arithmetic, the algebra is oriented toward teaching the reader a set of practical methods. In contrast to the direct instructions of the algorism, he explains his algebraic methods through specific problems, such as “half of a square and five roots are equal to twenty-eight dirhems [a unit of currency].”62 That is, in modern notation, x²/2 + 5x = 28. His solution for this problem begins as follows: “Your first business must be to complete your square, so that it amounts to one whole square. This you effect by doubling it. Therefore double it, and double also that which is added to it, as well as what is equal to it. Then you have a square and ten roots, equal to fifty-six dirhems.”63 What he is instructing us to do here—doubling the coefficients—is specific to this instance of the problem: if the first number is something other than a half, one has to multiply by some other number. Yet this example is meant to stand in for a broader class of problems. “Proceed in this manner,” he concludes the section, “whenever you meet with squares and roots that are equal to simple numbers: for it will always answer.”64 Al-Khwārizmī’s compiling of these instructions marks the beginning of what Victor J. Katz and Karen Hunger Parshall call the “algorithmic stage” in the history of algebra—a stage in which algebra consisted primarily of procedures for solving particular classes of problem.65

与算法一样，代数在欧洲长期以来被视为实用事物。花剌子密本人没有充分的理由将他的工作归类为实用或理论工作——阿巴斯王朝认为，伊斯兰教鼓励世俗知识和神圣知识之间的连续性。66欧洲大学的声望等级制度相对严格，代数显然不配获得学科地位。与算术和几何不同，代数并不在七门文科之中；它在欧洲的传播很大程度上要归功于商人，例如比萨的列奥纳多 (斐波那契)，他在陪同父亲前往北非进行贸易探险时接触了阿拉伯数学，并在 1202 年出版的《计算之书》中描述了代数方法。67文艺复兴时期，被称为“珠算学校”的意大利商业学院除了教授一系列计算实践外，还教授代数。68虽然大学有时会教授代数，但它的学术地位长期以来一直不如几何和算术那么安全。69即使在莱布尼茨的一生中，像艾萨克·巴罗（Isaac Barrow）这样的重量级人物（他于 1663 年成为剑桥大学第一位卢卡斯数学教授）也认为代数仅仅是一种缺乏论证力量的解决问题的技术。70

Like algorism, algebra was long regarded as a practical matter in Europe. Al-Khwārizmī himself would have had no strong reason to classify his work as either practical or theoretical—Islam, as it was interpreted by the Abbasids, encouraged a continuum between secular and holy knowledge.66 European universities had comparatively rigid hierarchies of prestige, and algebra did not self-evidently deserve the status of a learned discipline. Unlike arithmetic and geometry, algebra had no place among the seven liberal arts; its diffusion in Europe was largely due to merchants such as Leonardo of Pisa (Fibonacci), who encountered Arabic mathematics while accompanying his father on a trading expedition to North Africa and described algebraic methods in his 1202 Book of Calculation.67 During the Renaissance, the Italian mercantile academies known as “abacus schools” taught algebra alongside a range of calculating practices.68 While algebra was sometimes taught at universities, its academic status long remained less secure than those of geometry and arithmetic.69 Even in Leibniz’s lifetime, powerful figures such as Isaac Barrow, who became the first Lucasian professor of mathematics at Cambridge at 1663, were dismissing algebra as a mere problem-solving technique lacking the force of demonstration.70

代数最初缺乏声望，不仅是因为它与商业的联系，也是因为数字观念的冲突。在近代早期，欧洲数学深受欧几里得以及阿基米德和柏拉图等其他古希腊思想家的影响。71我们现在倾向于将基数视为一个单一概念，而欧几里得则使用了两个他认为完全不同的类似数字的概念：数量和量级。对欧几里得来说，数量是事物的数量，例如四个苹果，而量级是长度、面积或体积。欧几里得的量级也可以通过比率进行比较，他认为比率与量级本身不同；这种区别在使用不同的符号a / b = c / d和a : b :: c : d 时仍然存在。欧几里得还通过维度来区分量，而代数在使用平方数时违反了这一点：表达式x ² + x似乎是在长度上加了一个面积，用欧几里得的术语来说，这是无稽之谈。72

Algebra’s initial lack of prestige resulted not just from its association with commerce but also from conflicting ideas of number. In the early modern period, European mathematics was heavily under the sway of Euclid, along with other classical Greek thinkers such as Archimedes and Plato.71 Whereas we now tend to think of cardinal number as a singular concept, Euclid employs two number-like concepts that he treats as entirely distinct: quantity and magnitude. A quantity, for Euclid, is a number of things, such as four apples, whereas a magnitude is a length, area, or volume. Euclidean magnitudes can also be compared by means of ratios, which he treats as distinct from the magnitudes themselves; this distinction persists in the use of the different notations a/b = c/d and a : b :: c : d. Euclid also differentiated magnitudes by dimension, which algebra violated in its use of square numbers: the expression x² + x would seem to add an area to a length, which, in Euclidean terms, is nonsense.72

阿拉伯数学家可以广泛地接触希腊文献，因此他们的工作不应被视为完全独立的传统。但al-jabr和al-muqābala在某些方面确实与欧几里得的数字概念相冲突。解二次方程时，必须取平方根。当根为有理数时，情况很清楚：= 4。然而，也可能得到一个无法确定确切值的根。在《几何原本》第 10 卷中，欧几里得证明了存在长度不能表示为任何常见单位倍数的直线。73例如，正方形对角线与其中一条边的比值为，其值不完全等于任何分数。这样的数字在解二次方程时经常出现，并给“计算”精确数值解的尝试带来了问题。

Arabic mathematicians had extensive access to Greek sources, so their work should not be seen as a wholly separate tradition. But the al-jabr and al-muqābala did in some ways clash with Euclidean ideas of number. When solving a quadratic equation, one has to take square roots. The situation was clear enough when the root was rational: = 4. Yet one could also end up with a root whose exact value cannot be pinned down. In book 10 of the Elements, Euclid proves the existence of lines whose lengths cannot be expressed as multiples of any common unit.73 The proportion of a square’s diagonal to one of its sides, for instance, is , whose value is not quite equal to any fraction. Such numbers arose frequently in solving quadratic equations and posed a problem for attempts to “calculate” an exact numerical solution.

欧几里得为这些数字提供了一个暗示性的术语：alogos ( άλογος )，或者可以翻译为“无法言喻的”。74该术语表明，如果两个量之间的比率是无理的，那么它的真实值就无法说出。这种缺乏语言暗示的说法在中世纪和近代早期广为人知。虽然包括波斯天文学家 Jamshīd al-Kāshī 在内的许多思想家都开发了计算根的方法，但在这种情况下永远无法精确地做到这一点。75花拉子密代数书中的边注指出，人们必须满足于“近似值，而不是确切的真相：因为只有上帝知道确切的根是什么”。76这种态度在伊斯兰数学家以及一些基督徒中很常见，例如库萨的尼古拉和雅克·佩莱蒂埃，他们都将数学与神圣的神秘概念联系起来。77然而，其他人则对无理量的明显不可言喻性感到更加不安，无理量在代数中的使用似乎不符合欧几里得或阿基米德的知识标准。

Euclid provided a suggestive term for these numbers: alogos (άλογος), or, as it might be translated, unspeakable.74 If the ratio between two magnitudes is irrational, this term suggests, then its true value cannot be said. This suggestion of a lack of speech was widely known in the medieval and early modern periods. While a number of thinkers, including the Persian astronomer Jamshīd al-Kāshī, had developed methods for calculating roots, this could never be done exactly in such cases.75 A marginal note in a copy of al-Khwārizmī’s algebra book states that one must be content with “an approximation, and not the exact truth: for God alone knows what the exact root is.”76 Such attitudes were common among Islamic mathematicians as well as some Christians such as Nicolas of Cusa and Jacques Peletier, who both connected mathematics to a notion of divine mystery.77 Others, however, were more disturbed by the apparent unspeakability of irrational quantities, whose use in algebra did not seem to measure up to Euclidean or Archimedean standards of knowledge.

一种潜在的解决办法是忘记说出数值，而满足于说“二的平方根”。迈克尔·斯蒂费尔 (Michael Stifel) 于 1544 年出版的《积分算术》 (Arithmetica integra) 一书中出现了一个虽然模棱两可但却很有启发性的例子。78 斯蒂费尔写道：“关于无理数，人们有理由争论它们是真数还是虚构的。”79支持无理数存在的理由是它们在计算中很有用，因此“我们被感动并不得不承认，它们确实存在，也就是说，通过它们的影响，我们认为这些影响是真实的、确定的和恒定的。” 80另一方面，当我们“试图对它们进行计数，并用有理数对它们进行比例计算时，我们发现它们永远都会消失。” 81数字“消失”的想法（一个微妙而暴力的比喻）体现了新教徒对清晰理解作为数学真理标准的典型强调。斯蒂费尔是马丁·路德的私人朋友，曾因大胆表达支持而多次入狱，他并不满足于凭借权威来处理问题；他想要的是人们可以感知其力量的知识。82

One potential resolution was to forget about speaking the numerical value and be satisfied with saying “the square root of two.” A suggestive, albeit equivocal, example appears in Michael Stifel’s 1544 book Arithmetica integra.78 “It is justly disputed of irrational numbers,” Stifel writes, “whether they are true numbers, or fictions.”79 In favor of the existence of irrationals is their utility in calculation, on account of which “we are moved and compelled to confess, that they truly exist, namely by their effects, which we perceive to be real, certain, and constant.”80 On the other hand, when we “try to subject them to numeration, and proportion them by rational numbers, we find that they flee perpetually.”81 The idea that the numbers “flee” (a subtly violent metaphor) exemplifies a characteristically Protestant emphasis on clear apprehension as a standard of mathematical truth. A personal friend of Martin Luther who was repeatedly imprisoned for his bold expressions of support, Stifel was not content to take matters on authority; he wanted knowledge whose force one could perceive.82

尽管有这些保留意见，斯蒂费尔还是开发了一种方法，他称之为“算法”（Algorithmus ），用于使用有点类似现代的符号来处理无理数。83例如，在现代符号中，要计算，首先要确定两个根之间的比率，在本例中该比率是有理数：。那么，根的和必须与 成比例， 即 3 + 2 比 2；因此，我们现在写的和等于。基于此，斯蒂费尔确定。84他还包含一种更高级的“算法”，用于处理不同类型数字的复合数，例如。85这项工作（源于欧几里得对“无法言喻的”数字的讨论）表明，算法一词在16 世纪就已开始从其最初指的印地语 - 阿拉伯计算方法扩展到更广泛的符号方法类别。在这种情况下，值得注意的是，所讨论的方法处理的是无理数的精确关系，而不是数值近似值——这一特性使它们更接近代数而不是计算。

In spite of these reservations, Stifel developed a method, which he calls an “algorithm” (Algorithmus), for manipulating irrationals using a notation somewhat like the modern .83 For instance, in modern notation, to compute , one first determines the ratio between the two roots, which in this case turns out to be rational: . The sum of the roots, then, must stand in proportion to as 3 + 2 is to 2; hence, the sum, as we would now write it, is equal to . Based on this, Stifel determines that .84 He also includes a more advanced “algorithm” for working with composites of different types of number, such as .85 This work (which has roots in Euclid’s discussion of “unspeakable” numbers) indicates that the term algorithm was already, in the sixteenth century, beginning to expand beyond its original reference to the Hindi–Arabic methods of computation into a broader category of symbolic method. In this case, it is notable that the methods in question deal with exact relations of irrationals, not numerical approximations—a quality that aligns them more with algebra than with computation.

正如斯蒂费尔的例子所示，到了16世纪，欧洲数学家们开始感受到长期统治大学的古典希腊数论的局限性。几十年后，佛兰德数学家西蒙·斯蒂文（Simon Stevin）的著作首次明确批判了这些理论。1585年，斯蒂文发表了一篇这本小册子的标题被翻译成《十分之一》或《什一税》，他在其中将十进制表示法（其版本在中东和中国已经为人所知）引入了欧洲。86他没有使用小数点，而是使用圈出的数字来表示每个数字的意义：表示十分之三、百分之七、千分之五和万分之九，相当于我们今天写的 0.3759。87他接着表明，可以使用一种非常类似于普通算法的方法来对这些数字进行运算；他提供了在测量、测量和重商主义中的应用。

As the example of Stifel indicates, European mathematicians were, by the sixteenth century, feeling the limitations of the classical Greek number theories that had long reigned in universities. One of the earliest explicit repudiations of these theories appeared a few decades later in the work of the Flemish mathematician Simon Stevin. In 1585, Stevin published a pamphlet whose title is variously translated as The Tenth or The Tithe, in which he introduced decimal notation (versions of which were already known in the Middle East and in China) to Europe.86 Rather than a decimal point, he used circled numbers to indicate the significance of each digit: means three tenths, seven hundredths, five thousandths, and nine ten-thousandths, corresponding to what we would write today as 0.3759.87 He goes on to show that a method much like common algorism may be used to perform operations with these numbers; he provides applications in surveying, measurement, and mercantilism.

尽管与现代小数点相比，斯蒂文的圈出数字可能显得繁琐，但他的工作导致了数字概念的重大转变。对斯蒂文来说，数字的构成不是通过测量线或计算不可分割的单位，而是通过数字本身——那些运用算法的普通人的工具——来构建的。因此，斯蒂文坚持认为“不存在荒谬的、无理的、不规则的、无法解释的或无理数”。88人们可能无法将其写成分数，但可以（用现代符号）将其写成 1.41421356……并根据需要继续展开。1594 年，斯蒂文描述了一种可以做到这一点的方法：通过逐步将数轴分成十分之一，逐位确定等式的根。89（几个世纪后，图灵证明了“不可计算”数的存在，正如格雷戈里·柴廷所说，这些数字无法通过任何明确定义的程序生成；斯蒂文并不像他所想的那样站得住脚。90 ）虽然对于斯蒂文数的确切性质存在一些争论，但他的工作指向了现在所谓的实数连续统，这是一个与古典理论完全不同的数字概念。

Although Stevin’s circled numbers may have been cumbersome compared to the modern decimal point, his work led to a major shift in conceptions of number. For Stevin, a number is constructed not by measuring lines or counting indivisible units, but through the digits themselves—the tools of the commoners who practiced algorism. As a result, Stevin maintains that “there are no absurd, irrational, irregular, inexplicable, or surd numbers.”88 Perhaps one cannot write as a fraction, but one can write it (in modern notation) as 1.41421356 . . . and continue the expansion as far as one likes. In 1594, Stevin described a procedure that can do just that: pinning down the root of an equation digit by digit by progressively dividing the number line into tenths.89 (Centuries later, Turing would prove the existence of “uncomputable” numbers, as Gregory Chaitin put it, whose digits cannot be generated through any clearly defined procedure; Stevin was not on as steady ground as he thought.90) While there is some debate about the exact nature of Stevin’s numbers, his work points in the direction of what is now called the real number continuum, a number concept that breaks entirely with classical theories.

然而，斯蒂文对数的重新定义并未涵盖代数能够产生的所有数值实体。像斯蒂文这样的程序可以近似计算正数的根，包括诸如 这样的复合数，但它无法解释负数的根，而负数的根可以通过代数方法轻易得出。例如，如果将花拉子密的方程求解方法应用于我们现在写为x ² + (10 – x ) ² = 48 的方程，则可得到。91与无理数不同，负一的平方根甚至不能用小数近似，因为没有正数或负数具有负的平方。这个结果可以简单地被视为方程无解的标志而被拒绝，并且几个世纪以来这一直是典型的反应。然而，代数学程序能够产生这样的结果这一事实似乎意味着程序本身是没有根据的，而且随着数学家试图扩展其范围，问题变得更加严重。

Stevin’s redefinition of number does not, however, encompass every numerical entity algebra can produce. A procedure like Stevin’s can approximate the roots of positive numbers, including compounds such as , but it cannot account for the roots of negatives, which can readily emerge from algebraic methods. If, for instance, one were to apply al-Khwārizmī’s equation-solving method to the equation we would now write as x² + (10 – x)² = 48, one gets .91 Unlike irrationals, the square root of negative one cannot even be approximated by decimal fractions, since no positive or negative number has a negative square. This result could simply be rejected as a sign that the equation has no solution, and for centuries this would remain the typical reaction. Yet the fact that the procedures of algebra could produce such results seemed to imply that the procedures themselves were ill founded, and the problems worsened as mathematicians attempted to extend their range.

这一问题最著名的例子出现在意大利博学者杰罗拉莫·卡尔达诺的著作中。卡尔达诺在他 1545 年出版的《伟大的艺术，或代数规则》一书中描述了三次方程的完整解 - 即，用现代符号表示，形式为ax ³ + bx ² + cx + d = 0 的方程。对这种解决方案的渴望由来已久；波斯数学家奥玛·海亚姆（最著名的是诗歌《鲁拜集》的假定作者）在 11 世纪分析过三次方程，但无论是他还是他的直接追随者都无法找到算法解。92卡尔达诺的解决方案引发了一场众所周知的丑闻，让人们得以一窥 16 世纪意大利数学家的生活。卡尔达诺承认，他从一位名叫塔尔塔利亚（意思是“口吃者”）的熟人分享的一首未发表的诗中了解到了部分解决方案。塔尔塔利亚要他发誓保守这个结果的秘密，但卡尔达诺还是发表了。卡尔达诺有个借口。他发现，塔尔塔利亚并不是第一个发现这个结果的人；另一位名叫西皮奥内·德尔·费罗的数学家十多年前就发现了它。卡尔达诺认为这意味着可以公平地发表结果。然而，他没有承认誓言，这导致了一场激烈的争论，最终导致了一场文艺复兴式的决斗替代方案：塔尔塔利亚和他的学生卢多维科·费拉里之间的公开数学竞赛，卢多维科·费拉里战胜了卡尔达诺，结束了这场争论。

The best-known instance of this issue appears in the work of the Italian polymath Gerolamo Cardano. In his 1545 book The Great Art, or, the Rules of Algebra, Cardano described a complete solution for cubic equations—that is, in modern notation, equations of the form ax³ + bx² + cx + d = 0. The desire for such a solution had been long-standing; the Persian mathematician Omar Khayyam (best known as the presumed author of the poetic cycle the Rubaiyat) had analyzed cubic equations in the eleventh century, but neither he nor his immediate followers could find an algorithmic solution.92 Cardano’s solution was attended by a well-known scandal that gives a taste of what life as a mathematician was like in sixteenth-century Italy. Cardano learned part of the solution, as he acknowledges, from an unpublished poem shared by an acquaintance known as Tartaglia (which means “the stammerer”). Tartaglia swore him to secrecy about this result, and yet Cardano published it anyway. Cardano had an excuse. Tartaglia, he had discovered, was not the first to discover the result; another mathematician named Scipione del Ferro had discovered it over a decade before. Cardano took this to mean it was fair game to publish. Yet he failed to acknowledge the oath, which led to a bitter dispute culminating in a Renaissance alternative to a duel: a public mathematics contest between Tartaglia and one of Cardano’s students, Ludovico Ferrari, who bested the stammerer and put the matter to an end.

卡尔达诺提出的解决方案将问题分解成更具体的情形，例如x ³ + ax = b和x ³ = ax ² + b。这种分解是必要的，因为与花剌子密一样，卡尔达诺不允许方程式出现负系数；因此不可能将所有三次方程式合并成一个通用形式。他的“规则”以晦涩难懂的散文形式呈现，偶尔会使用缩写，例如用℞表示“根”。例如，要解x ³ + ax = b，请遵循以下说明：

The solution Cardano assembled breaks the problem down into more specific cases such as x³ + ax = b and x³ = ax² + b. This division into cases is necessary because, like al-Khwārizmī, Cardano does not allow equations to have negative coefficients; it is thus impossible to combine all cubic equations into one general form. His “rules” take the form of knotty prose with the occasional use of an abbreviation, such as ℞ for “root.” To solve x³ + ax = b, for instance, one follows these instructions:

将事物数量系数的三分之一立方，加上等式中常数的一半的平方；然后对整体取平方根。重复此过程，将两个数中的一个加上已求平方的数的一半，再从另一个数中减去相同的数。这样你就得到了一个二项式及其极值。然后，用二项式的立方℞减去极值立方℞，余数就是事物的价值。93

Cube one-third the coefficient of the number of things, add it to the square of one-half the constant of the equation; & take the square root of the whole. You will duplicate this, and to one of the two you add the one-half of a number you have already squared and from the other you subtract the same. You will then have a binomium and its apotome. Then, subtracting the cube ℞ of the apotome from the cube ℞ of the binomium, the remainder [or] that which is left is the value of the thing.93

数学史学家海伦娜·M·皮乔尔 (Helena M. Pycior) 将这些程序描述为“散文算法”，它们很容易被解释为现代意义上的算法。94与花拉子米和斯蒂费尔不同，卡尔达诺卡尔达诺以通用形式而非具体示例来呈现规则，预示了后来成为编程语言特征的抽象概念。但卡尔达诺的理论并非完全是算法性的。除了每条“规则”，他还提供了其有效性的几何证明，并附上证明，“以便除了单纯的实验知识之外，推理可以强化对结果的信念”。95因此，他的书的目标不仅仅是汇编解决实际问题的程序；读者也应该理解这些程序的正确性。

The historian of mathematics Helena M. Pycior characterizes these procedures as “prose algorithms,” and they can readily be interpreted as algorithms in the modern sense.94 Unlike al-Khwārizmī and Stifel, Cardano presents the rules in general forms rather than by means of specific examples, prefiguring the abstraction that would later come to characterize the programming language. But Cardano’s theory is not wholly algorithmic. Along with each “rule,” he presents a geometric demonstration of its validity, which he includes “so that, beyond mere experimental knowledge, reasoning may reinforce belief” in the results.95 The goal of his book, then, was not just to compile procedures for solving practical problems; the reader was also supposed to come away with an understanding of why those procedures were right.

至少，理想情况是这样的。卡尔达诺方法的几何基础在所谓的不可约情形下遇到了麻烦。正如皮乔尔指出的那样，某些三次方程，例如看似无害的y ³ = 8 y + 3，存在有理解，而用卡尔达诺的方法无法找到这些解，除非遇到负数的平方根。96在这种情况下，应用卡尔达诺规则可以得到（用现代符号表示的）这个相当繁琐的表达式：

Such, at least, was the ideal. The geometric basis of Cardano’s methods ran into trouble in the so-called irreducible case. As Pycior points out, certain cubic equations, such as the innocent-looking y³ = 8y + 3, have rational solutions that one cannot find via Cardano’s methods without encountering the square roots of negatives.96 In this case, applying Cardano’s rule gives (in modern notation) this rather cumbersome expression:

使用类似斯蒂费尔的“复合无理数算法”之类的程序，可以将该值简化为y = 3，这显然满足原始方程。97按照21世纪代数的标准，这个结果是正确的，但16世纪的大多数数学家认为这样的表达式毫无意义。卡尔达诺的一些早期追随者，例如拉斐尔·邦贝利，设法克服了他们的保留意见；邦贝利写道，负数的平方根最初“在我看来更多的是基于诡辩而不是真理，但我一直寻找，直到找到证明” 。98然而，此类证明的理论基础仍然高度不确定。

Using a procedure like Stifel’s “algorithm of composite irrational numbers,” one can reduce this value to y = 3, which clearly does satisfy the original equation.97 This result is correct by the standards of twenty-first-century algebra, but the majority of mathematicians in the sixteenth century viewed such expressions as nonsense. Some of Cardano’s early followers, such as Raphael Bombelli, managed to overcome their reservations; square roots of negative numbers, Bombelli wrote, initially “seemed to me to be based more on sophism than on truth, but I searched until I found the proof.”98 Yet the theoretical basis for such proofs remained highly uncertain.

需要强调的是，卡尔达诺对数字新领域的尝试并不依赖于符号的使用。卡尔达诺无疑在进行代数推理，但除了数字和一些缩写外，他还用文字解释他的程序。代数的这一方面很快就会发生变化。早在 16 世纪中期，中世纪方程求解程序的复杂散文就开始让位于紧凑的公式。到了 17 世纪中期，符号占据了主导地位。正如我在下一节中讨论的那样，这种新符号的大部分威力来自于字母的使用：x、y、z表示未知值，a、b、c表示已知值。与具有固定含义的数字不同，这些符号在每次求解时都会具有不同的值。乔纳斯·摩尔 (Jonas Moore)在其 1650 年出版的教科书《算术》中这样解释了它们之间的区别：数字是“确定的、确定的符号”，而代数符号是“不确定的、不确定的、任意的符号” 。99 矛盾的是，在许多思想家看来，这些“不确定”的符号却达到了文字无法比拟的清晰度——这一发展几乎立即激发了人们对一种通用方法的梦想。

Cardano’s tentative foray into new realms of number, it should be emphasized, did not depend on the use of symbols. Cardano was undeniably practicing algebraic reasoning, but apart from numerals and a few abbreviations, he explained his procedures in words. This aspect of algebra was soon to change. Already in the mid-1500s, the convoluted prose of medieval equation-solving procedures was starting to give way to compact formulae. By the mid-1600s, symbols had taken over. As I discuss in the next section, this new notation gained much of its power from the use of letters: x, y, z for unknown values and a, b, c for known ones. Unlike numerals, with their fixed meanings, these symbols bear different values with every problem solved. In his 1650 textbook Arithmetick, Jonas Moore explains the difference this way: numerals are “Notation certain, and determinate,” whereas algebraic symbols are “Notation uncertain, undeterminate, and arbitrary.”99 These “uncertain” symbols seemed to many thinkers, paradoxically, to achieve a level of clarity words could not match—a development that almost immediately inspired dreams of a universal method.

从数字到字母

From Numbers to Letters

在当今的小学，从算术到代数的过渡以字母的突然出现为标志。在学习了多年诸如 4 + 5 之类的固定值之后，学生突然面临诸如 4 + x之类的表达式，并且必须学习如何推理尚未知的值。对代数来说，需要某种方式来表示未知量，但使用字母来填补这一空缺是相对较新的发展。100现代代数符号基于一个后来成为编程语言设计核心的思想——使用任意符号来表示未指定的值，以便概括某个过程。这种符号的使用已经深深植根于现代算法思维中，但在花剌子密本人的著作中却没有，而且它的出现使得计算过程与意义的关系变得错综复杂。

In today’s primary schools, the transition from arithmetic to algebra is marked by the sudden appearance of letters. After years of learning about fixed values such as 4 + 5, the student is suddenly confronted with expressions such as 4 + x and must learn how to reason about a value that is not yet known. The need for some way of referring to unknown quantities is essential to algebra, but the use of letters to fill this office is a relatively recent development.100 Modern algebraic notation is based on an idea that would later become central to the design of programming languages—the use of arbitrary symbols to represent values that are left unspecified for the purpose of generalizing a procedure. This use of symbols has become deeply entrenched in modern algorithmic thinking, but it was not there in the work of al-Khwārizmī himself, and it came at the cost of placing computational procedures in a fraught relation to meaning.

早期阿拉伯代数迭代最常用词语表示未知量。中世纪阿拉伯作家将未知数称为shay ，意为事物；在拉丁语中，它变成了res，卡尔达诺也使用这个词。但早在卡尔达诺时代之前，就已经出现了表达未知数的其他方式。罗伯特·切斯特于 1145 年左右翻译的《算术集》，其中简要概括了算术“规则”（regul[a]e），其中使用了阿拉伯语原文中没有的符号。巴拉巴斯·B·休斯在他的评注版中将这些符号近似为ø、ϑ和ʒ。例如：“当ø等于ϑ和ʒ 时，ø和ϑ必须除以ʒ，ϑ对半切入自身，将乘积加到数字上。聚合整数减去半ϑ的基数即为所求值。” 101这些字母对应于特定次数项的系数：ø为常数，ϑ为未知数，ʒ 为未知数的平方。因此，用现代术语来说，该方程为ʒ x ² + ϑ x = ø。

Early iterations of Arabic algebra most commonly represented unknown quantities with words. Medieval Arabic writers referred to the unknown as shay meaning thing; in Latin, this became res, which is the word Cardano uses. But long before Cardano’s time, other ways of expressing unknowns had appeared. Robert of Chester’s c. 1145 Latin translation of the Al-jabr includes a condensed summary of the “rules” (regul[a]e) of the art that uses symbols not in the Arabic original. In his critical edition, Barabas B. Hughes approximates these symbols as ø, ϑ, and ʒ. For instance: “When ø is equal to ϑ and ʒ, ø and ϑ must be divided by ʒ, ϑ halved, the half drawn into itself, the product added to the number. The radix of the aggregated whole minus half ϑ reveals what is sought.”101 The letters correspond to coefficients attached to terms of specific degrees: ø is a constant, ϑ the unknown, ʒ the square of the unknown. In modern terms, then, the equation is ʒx² + ϑx = ø.

然而，将切斯特的符号与那样的现代符号相提并论是误导。他的ø、ϑ和ʒ并不是像现代代数中的a和b那样可互换的标记；它们的功能更像是普通名词，借用戈特洛布·弗雷格的一对术语来说，它们既有义项，又有指称。102用弗雷格的话来说，指称就是被单独挑出来的东西：例如，短语“说谎者”的指称就是被称为说谎者的人。义项是词语所承载的语义负荷：在这种情况下，就是“做一个说谎者”。切斯特的符号运用了这两种含义，既指数字，又指代数字。系数的值，并传达有关这些值如何融入方程的信息 - 即哪个值应用于根，哪个值应用于根的平方。当然，现代代数的符号中有一些是有意义的；勒内·笛卡尔制定了使用x，y和z表示未知数，使用a，b和c表示已知数的惯例，从而使用字母作为不同值用途的提示。但符号操作规则并不依赖于这些惯例；a和x遵循相同的规则。另一方面，如果不认识符号的不同含义，就无法理解切斯特的“规则”。

Yet placing Chester’s symbols in the company of modern notation like that is misleading. His ø, ϑ, and ʒ are not interchangeable tokens like the as and bs of modern algebra; they function more like common nouns in that they have, to borrow a pair of terms from Gottlob Frege, both sense and reference.102 In Frege’s terms, reference is what is singled out: the reference of the phrase the liar, for instance, is the person being called a liar. Sense is the semantic freight the words carry: in this case, all it is to be a liar. Chester’s symbols employ both types of meaning, at once referring to numerical values of the coefficients and conveying information about how those values fit into the equation—about, that is, which value is applied to the root and which to the square of the root. There is, to be sure, something of sense in the symbols of modern algebra; René Descartes instated a convention of using x, y, and z for unknowns and a, b, and c for knowns, thus using the letters as hints as to the purposes different values play. But the rules of symbol manipulation do not depend on these conventions; a and x obey identical rules. Chester’s “rule,” on the other hand, is incomprehensible without a recognition of the different senses of the symbols.

从切斯特的ø、ϑ和ʒ到我们的a和x的转变并不是一个线性过程。在 1494 年出版的《算术大全》中，卢卡·帕乔利 (Luca Pacioli) 提出了一种简洁的方程式，将未知数缩写为“co.”，取自意大利语cosa（事物）。103这个词启发了一种后来被称为“cossic 字符”的符号，它放在数字旁边，表示未知数的一定幂。这些符号最早出现在克里斯托夫·鲁道夫 (Christoff Rudolff) 1525 年左右的代数教科书中，该书的缩写书名Die Coss让一代德国人用一个在词源上意为“事物”的术语来指代代数。104后来，Stifel 和 Recorde 采用了鲁道夫的符号，而其他人则采用了类似的符号，但符号不同（图 1.3）。切斯特符号表示未指定的系数，而经典符号则伴随系数来表示项的度数。因此，从某种意义上说，这些符号不如切斯特的ø、ϑ和ʒ抽象，它们为未知数提供了符号，但没有提供任何方法使问题的参数保持未指定状态，从而可以将解决方案表述为一般规则。

The transition from Chester’s ø, ϑ, and ʒ to our as and xs was not a linear process. In his 1494 Summa de arithmetica, Luca Pacioli presented equations in a compact form that abbreviated unknowns as “co.,” after the Italian cosa (thing).103 This word inspired a type of symbol that came to be known as the “cossic character,” which was placed next to a number to indicate a certain power of the unknown. These symbols first appeared in Christoff Rudolff’s c. 1525 algebra textbook, the shortened title of which—Die Coss—led a generation of Germans to refer to algebra with a term that means, etymologically, “the thing.”104 Rudolff’s symbols were later adopted by Stifel and Recorde, while others employed similar notations with different symbols (figure 1.3). Whereas Chester’s symbols represent unspecified coefficients, cossic symbols accompany coefficients to indicate the degree of the term. The symbols are thus, in a sense, less abstract than Chester’s ø, ϑ, and ʒ, providing a notation for the unknown but offering no way to leave the parameters of a problem unspecified so that the solution can be stated as a general rule.

与 Recorde 的 = 符号相结合，宇宙符号使得构建有时被称为有史以来第一个方程式的东西成为可能（图 1.4 ）。由于 Recorde 版本的宇宙符号中没有文字，Heeffer 认为他的代数是完全符号化的。105但从语义的角度来看，Recorde 的符号与我们习惯的符号截然不同。在现代符号中，图中第一个方程的左边是 14 x + 15——一个在x确定之前其值不确定的表达式。这种不确定性与 Recorde 的代数不同。106相反，Recorde 将宇宙符号解释为测量单位——人们谈论五个根或五个平方就像谈论“20. shippes”一样。107与普通算法一样，宇宙字符具有“计数法”，需要将它们转换成文字——在这种情况下，我们可能会说类似14 个根号加 15 个数。108这些词表达的不是不确定的数量，而是“复合数”，就 Recorde 而言，它具有与12一样的固定值。109虽然他使用符号，但 Recorde 仍然主要关注表示数字，他将其理解为事物的数字。

Combined with Recorde’s = sign, cossic characters enabled the construction of what are sometimes characterized as the first-ever equations (figure 1.4). On account of the absence of words in Recorde’s version of cossic notation, Heeffer argues that his algebra was fully symbolic.105 But Recorde’s symbols are, from a semantic perspective, quite different from the ones we are used to. In modern notation, the left-hand side of the first equation in the figure would be 14x + 15—an expression whose value is indeterminate until x is fixed. This indeterminacy is alien to Recorde’s algebra.106 Instead, Recorde explains cossic characters as units of measurement—one talks about five roots or five squares in the same way one talks about “20. shippes.”107 As in common algorism, the cossic characters have a “numeration” that involves translating them into words—in this case, we might say something like fourteen roots more fifteen nombers.108 These words express not an indeterminate quantity but rather a “compounde nomber” that has, as far as Recorde is concerned, a fixed value as much as twelve does.109 While he employed symbols, Recorde was still concerned primarily with representing numbers, which he understood as numbers of things.

就算法思维而言，一个关键的转折点发生在法国律师、亨利三世和四世国王的顾问、业余天文学家弗朗索瓦·维特（François Viète）的著作中。在他1591年出版的《艺术导论》中， 分析中，韦达引入了一种保留单位但取消了数字的符号：

As far as algorithmic thinking goes, a crucial turning point occurred in the work of the French lawyer, councillor to Kings Henri III and IV, and amateur astronomer François Viète. In his 1591 Introduction to the Art of Analysis, Viète introduced a notation that kept the units but did away with numbers:

用现代符号表示，则是。110但这些符号并不完全相同。“平面”指定A是二维的，这是必要的，因为对韦达而言，长度和面积是不同类型的值，不能加在一起。111如果这些维度表明对古典数字观念的依恋，那么韦达对字母的运用则指向了一个新的方向。与古典符号相反，韦达使用字母表示未知数和已知数。他用元音表示未知数，用辅音表示已知数，正如他所写的，“这样这项工作就可以借助某种技巧。” 112这些他称之为“种类”的字母使一种在英语中称为“似是而非的算术”的做法成为可能（specious 的发音可能与拉丁语中的硬 c 相同）。人们现在可以得出关于一般“种类”方程的结论，例如Ax + B = C，而不必研究诸如 14 x + 15 = 71 之类的单个方程。

This would be, in modern notation, .110 Yet the notations are not quite equivalent. The “planum” specifies that A is two-dimensional, which is necessary because, for Viète, lengths and areas are different types of value that may not be added together.111 If these dimensions indicate an attachment to classical ideas of number, Viète’s use of letters pointed in a new direction. In contrast to the cossic notations, Viète used letters to represent both the unknowns and the knowns. He used vowels to represent unknowns and consonants for knowns in order, as he wrote, “that this work may be assisted by some art.”112 These letters, which he called “species,” enabled a practice known in English as “specious arithmetic” (specious was probably pronounced with a hard c, as in Latin). Instead of working on single equations such as 14x + 15 = 71, one could now draw conclusions about general “species” of equation such as Ax + B = C.

关于韦达的A和B的来源，存在许多不同的说法，并且尚不清楚它们是否与鲁道夫和雷科德使用的经典字符有任何联系。113约翰·沃利斯 (John Wallis)在其 1685 年的《代数学论》中将这些字母与韦达的法律背景联系起来，认为它们源自律师缩写人名的方式。114雅各布·克莱因(Jacob Klein) 在其关于数字概念史的经典研究中，将它们视为全新事物，认为韦达开创了从古典数学概念向“符号形式主义”的转变。115杰弗里·奥克斯 (Jeffrey A. Oaks) 对韦达的最新解释是，他认为克莱因忽视了韦达方法的几何基础；在奥克斯的解释中，韦达借鉴了使用字母来指代几何图形元素的做法。116然而，有一个值得注意的区别。几何图形必然会以特定的比例来表示图形，即使该图形旨在代表一类通用的图形。人们可以画出许多比例各异的不等边三角形，但无法画出比例不确定的不等边三角形。另一方面，韦达符号使得人们能够抽象地描述这些过程，而无需任何特定的例子。

There have been a number of differing accounts of where Viète’s As and Bs came from, and it is not clear that they have any connection at all to the cossic characters used by Rudolff and Recorde.113 In his 1685 Treatise of Algebra, John Wallis linked the letters to Viète’s background in law, arguing that they originated from a way lawyers abbreviated people’s names.114 In his classic study of the history of number concepts, Jacob Klein treats them as wholly novel, arguing that Viète initiated the turn away from classical conceptions of mathematics toward “symbolic formalism.”115 The best recent account of Viète is by Jeffrey A. Oaks, who argues that Klein overlooked the geometric basis of Viète’s method; in Oaks’s account, Viète was drawing on the practice of using letters to refer to elements of geometric diagrams.116 There is, however, a difference worth noting. Of necessity, geometric diagrams show a figure with particular proportions, even when the drawing is meant to stand in for a general class of figure. One can draw many scalene triangles with various proportions, but one cannot draw a scalene triangle with indeterminate proportions. Viète’s notation, on the other hand, enables procedures to be described abstractly, without the need for any particular example.

韦达的思想是人文主义者努力恢复古代知识遗失形式的一部分。当时，人们对亚历山大的丢番图以及后来亚历山大的塞翁和帕普斯的三世纪著作的兴趣重新燃起。丢番图的著作被阿拉伯学者保存下来，但在欧洲并未被广泛阅读。直到 16 世纪 70 年代，拉斐尔·邦贝利 (Raphael Bombelli) 将其部分内容译成拉丁文。117丢番图可能借鉴了古希腊、埃及和巴比伦数学传统的结合，发展出了求解简单方程以及不定方程组（即未施加足够条件来排除无穷多解存在的方程组）的方法。118他的解法以类似宇宙字符的紧凑符号表示，涉及对值的关系做出任意假设，然后计算出它们的后果。119例如，为了解决求和为 16 的两个平方数的问题，他假设它们的根符合我们现在写的关系y = 2 x + 4。120基于这一假设，可解原方程得到该问题的解：和。

Viète’s idea emerged as a part of the humanist push to recover supposedly lost forms of ancient knowledge. There had, in particular, been a revival of interest in the third-century work of Diophantus of Alexandria, along with the later Alexandrians Theon and Pappus. Diophantus’s work was preserved by Arabic scholars, but it was not widely read in Europe until the 1570s, when portions of it were translated into Latin by Raphael Bombelli.117 Probably drawing on a combination of ancient Greek, Egyptian, and Babylonian mathematical traditions, Diophantus had developed methods for solving simple equations as well as indeterminate systems of equations—that is, systems that do not impose enough conditions to exclude the existence of infinitely many solutions.118 His solutions, which he presents in a compact notation somewhat like cossic characters, involve making arbitrary assumptions about the relations of values and then working out their consequences.119 For instance, to solve the problem of finding two squares that sum to 16, he assumed that their roots stood in the relation, as we would write it now, y = 2x + 4.120 On this assumption, the original equation may be solved to produce a solution to the problem: and .

不管公平与否（几乎可以肯定是公平的），韦达将自己定位为将方程式理论恢复到原始的亚历山大状态。他写道，这门艺术“已被野蛮人破坏和玷污”，因此他必须“摆脱所有伪技术术语（pseudo-categorematis），以免它保留污秽并继续散发旧臭”。121毫不掩饰的潜台词是，他希望抹去阿拉伯语资料，转而使用希腊语资料，这种沙文主义体现在他不喜欢“代数”一词上。122韦达希望用他认为是丢番图的秘密方法（他称之为分析）取代花剌子密的技术。123在韦达的定义中（他将此定义归功于亚历山大的泰翁），分析始于假设所寻求的内容并推导出其后果，而综合始于真实，并推导出其他真理。124韦达将分析分为三个阶段。第一阶段是阐释阶段，它涉及使用似是而非的算术，根据已知内容推导出未知内容的公式。第二阶段是阐释阶段，它涉及制定和证明由此得出的解。最后是注释阶段或修辞阶段，在此阶段“从建立的方程式或比例中可以得出所寻求的数量本身”。125换句话说，修辞分析是“设A = 5”的时刻：人们设定字母的值并将公式解析为数字的时刻。

Fairly or not (almost certainly not), Viète positioned himself as restoring the theory of equations to a pristine Alexandrian state. This art, he wrote, has been “spoiled and defiled by the barbarians,” on account of which he must get “rid of all its pseudo-technical terms (pseudo-categorematis) lest it should retain its filth and continue to stink in the old way.”121 The unsubtle subtext is that he wishes to expunge Arabic sources in favor of Hellenistic ones, a chauvinism manifest in his dislike of the term algebra.122 Viète hoped to replace al-Khwārizmī’s techniques with what he took to be Diophantus’s secret method, which he called analysis.123 In Viète’s definition, which he attributes to Theon of Alexandria, analysis starts by assuming what is sought and working out its consequences, whereas synthesis starts from what is true and deduces other truths.124 Viète divided analysis into three phases. The first is the zetetic, which involves the use of specious arithmetic to derive a formula for the unknown in terms of what is known. The second phase, the poristic, involves the formulation and proof of the resulting solution. Last is the exegetic or rhetic phase, in which “from the equation set up or the proportion there is produced the magnitude itself which is sought.”125 Rhetic analysis, in other words, is the “let A = 5” moment: the moment in which one sets the values of the letters and resolves the formula into a number.

在他后来的著作《凿壁学》（1593 年）中，韦达提出了大量已解决的问题，这些问题比《引言》中的一般解释更能说明这种方法实际上是如何运作的。例如，他考虑了这个问题：“给定线之差和立方差：求线。” 126也就是说，找出两个数，它们的差为B，立方差为D。韦达这样解决了这个问题。将这两个数的和记为E；那么E + B是较大数的两倍，而E − B是较小数的两倍。通过一些代数运算，他得到了这个结果：

In his later work Zetetics (1593), Viète presents a large number of worked-out problems that give a better sense of how this method actually worked than the general explanations of the Introduction. For instance, he considers this problem: “Given the difference of lines, & difference of cubes: to find the lines.”126 That is, find two numbers that differ by B, and whose cubes differ by D. Viète solves it like so. Call the sum of the numbers E; then E + B is twice the larger number and E − B is twice the smaller number. Through some algebraic manipulation, he ends up with this:

也就是说，就像我们现在写的一样，。接下来，他用文字重述了公式：“四倍立方差减去线差的立方，如果应用于三倍线差：则得出总线的平方。” 127最后，他代入值B = 6 和D solidum = 504 得到数值答案。事实上，他用一种与他用来处理方程的符号完全不同的符号来表示结果，这一点经常被人们忽略：“ summa laterum 1 N , 1 Q æquatur 100。” 128与古字符非常相似，这种符号将未知值视为测量单位。在这种情况下，N是一个设置为等于“线之和”的单位，Q是另一个等于其平方的单位；因此，和的平方是 100。由此，我们可以推断出和本身是 10，并且满足问题的两个数字是 2 和 8。129该方法与丢番图的方法相似，但有一点不同：丢番图始终使用数字，而韦达则使用字母完成大部分工作，只在末尾插入数字。

That is, as we would now write, . Next, he restates the formula in words: “The quadruple difference of cubes, minus the cube of the difference of lines, if applied to triple the difference of lines: appears the square of the aggregate line.”127 Finally, he plugs in the values B = 6 and D solidum = 504 to get a numerical answer. In a fact often glossed over, he presents the result in an entirely different notation from the one he used to manipulate equations: “summa laterum 1N, 1Q æquatur 100.”128 Much like cossic characters, this notation treats the unknown value as a unit of measurement. In this case, N is a unit set to equal the “sum of the lines” and Q is another unit equal to its square; thus, the square of the sum is 100. From this, we can deduce that the sum itself is 10, and two numbers that satisfy the problem are 2 and 8.129 The approach parallels that of Diophantus, but with a difference: whereas Diophantus uses numbers all along, Viète does most of the work with letters, only plugging in the numbers at the end.

韦达对这门艺术的抱负可谓宏大。他在1591年的《导言》中以大写字母的帝国主义宗旨声明结尾：“不留 任何 未 解决的问题 ” 。130这句话标志着——与现代代数符号主义的曙光同时出现的——对这些符号功能的普遍化追求的到来。韦达的符号法和编程语言之间存在着明显的共鸣，事实上，两者之间也存在着一条历史影响线。但这条道路同样不是直线性的。如上一段中的例子所示，韦达所做的并不完全是我们所知的符号代数。从现代的角度来看，公式的文字陈述似乎是多余的——如果已经有了符号，为什么还要用文字呢？他在修辞阶段代入数字来得到答案，这提出了一个更微妙的问题，但尤其能说明通用计算方案将面临的问题。动词“let ”（在拉丁语中，韦达使用了祈使语气虚拟式“sit”，这在几何证明中很常见）表示似是而非的算术中未确定的“B ”和“D ”与数字领域之间的中介关系。韦达对这种中介关系的自觉性显而易见：在他“赋予”字母值的同时，他立即切换到一种更符合古典数字观念的符号。因此，在修辞时刻，符号的性质发生了变化，不再是抽象的指令，而是开始代表事物的数字。

Viète was grandiose in his ambitions for this art. He ends the 1591 Introduction with an imperialistic statement of purpose set in caps: “TO LEAVE NO PROBLEM UNSOLVED.”130 This statement signals the arrival—right alongside the first glimmerings of modern algebraic symbolism—of universalizing ambitions for what those symbols could do. There is a clear resonance between Viète’s notation and programming languages and, indeed, a historical line of influence linking the two. But the road, again, is not linear. As the example in the foregoing paragraph shows, what Viète was doing was not quite symbolic algebra as we know it. The verbal statement of the formula seems, from a modern point of view, redundant—why bother with words if you already have the symbols? The rhetic phase, in which he plugs in the numbers to get an answer, raises an issue that is subtler but especially indicative of the problems that would come to face universal computation schemes. The verb let—in Latin, Viète uses the jussive subjunctive sit, as was common in geometric proofs—indicates a mediation between the undetermined Bs and Ds of specious arithmetic and the realm of numbers. Viète’s self-consciousness about this mediation is apparent from the fact that the instant he “lets” the letters have values, he switches to a different notation more aligned with classical ideas of number. At the rhetic moment, then, the symbols change in nature, ceasing to be abstract instructions and coming to represent numbers of things.

所有这些复杂性很快就会消失。虽然韦达的分析思想会产生持久的影响，但他作品中最重要的元素是字母，它们很快就被从其言语背景中抽离出来。最早、最热情的采用者之一是托马斯哈里奥特。131哈里奥特最出名的事迹是陪同沃尔特·雷利爵士前往罗阿诺克岛，并写了一本关于他在弗吉尼亚经历的书。他的代数著作直到 1631 年才出版，大约在他去世十年后，尽管它可能更早以手稿形式流传。132除了序言和几页拉丁文定义之外，1631 年的这本卷书几乎全部由符号组成。哈里奥特开发了一种简化版的韦达符号，删除了拉丁文单词：他把A cubum写成aaa（图 1.5）。该书的前言可能是沃尔特·沃纳撰写的，其中指出韦达使用“解释符号”进行分析，而哈里奥特发现使用“文字符号：即字母表中的字母”更方便。133哈里奥特还打破了韦达如此仔细地划分分析三个阶段的语言障碍。一旦数字进入场景；现在，相同的符号可以用于代数运算和数值计算。

All this complexity would soon vanish. While Viète’s idea of analysis would have a lasting influence, the most consequential element of his work was the letters, which were soon extracted from their verbal settings. Among their earliest and most enthusiastic adopters was Thomas Harriot.131 Harriot was best known for accompanying Sir Walter Raleigh to Roanoke Island and for writing a book about his experiences in Virginia. His algebraic work was not published until 1631, about a decade after his death, although it probably circulated in manuscript form earlier.132 After a preface and a few pages of definitions in Latin, the 1631 volume consists almost entirely of symbols. Harriot developed a simplified version of Viète’s notation that expunged the Latin words: in place of A cubum, he wrote aaa (figure 1.5). The preface to the book, probably written by Walter Warner, states that Viète had performed his analyses with “interpreted signs,” whereas Harriot found it more convenient to use “a literal notation: that is, the letters of the alphabet.”133 Harriot also broke down the linguistic barriers by which Viète had so carefully demarcated the three phases of analysis. No more would cossic notations reappear once numbers enter the scene; now, the same symbols could be used for both algebraic manipulation and numerical computation.

这些新符号的崇拜者们很快就想到将它们扩展到其他领域。十七世纪三十年代，约翰·佩尔试图将韦达和哈里奥特的思想发展成一套更完整、更严格的方程理论。134佩尔将对代数的兴趣与对速记等新型交流方式的迷恋联系起来，他的岳父亨利·雷诺兹对速记深感兴趣。135与当时许多其他人一样，佩尔认为似是而非的算术可以为一种新的书写形式提供模型，这种书写形式可以用符号表达任何事物。136这个想法是将概念分解成简单的组成部分——例如，他认为火结合了“热的东西”和“闪亮”——并为这些简单组成部分开发符号，使它们可以以各种配置组合在一起。137这种“组合艺术”的想法由来已久，但韦达对分析一词的使用提供了一种新的解释方法。将概念分解成部分是否真的与代数分析有关，长期以来一直是争论的话题。无论如何，佩尔的例子表明，韦达的早期追随者的兴趣不仅仅在于解决数值问题——他们还从他的方法中看到了一些可以应用于任何研究主题的东西。

It did not take long for admirers of these new symbols to conceive of extending them to other areas. In the 1630s, John Pell attempted to develop the ideas of Viète and Harriot into a more complete and rigorous theory of equations.134 Pell connected this interest in algebra to a fascination with novel forms of communication such as shorthand, in which his father-in-law, Henry Reynolds, was deeply interested.135 As did a number of others at the time, Pell thought specious arithmetic could provide a model for a new form of writing that could express anything in symbols.136 The idea was to divide concepts up into simple components—he thought fire, for instance, combined “hot thing” and “shining”—and develop symbols for those simple components that could be placed together in various configurations.137 The idea of such an “art of combinations” was very old, but Viète’s use of the term analysis provided a new way of explaining it. Whether the analysis of concepts into parts really had anything to do with algebraic analysis would long be a topic of debate. Regardless, the example of Pell shows that Viète’s early followers were interested in more than solving numerical problems—they also saw something in his method that could be applied to any number of topics of inquiry.

哈里奥特和佩尔都位于英国，17 世纪初，那里对符号的狂热最为强烈。在英吉利海峡对岸，皮埃尔·德·费马、马里诺·盖塔尔迪和让-路易·沃勒扎德将韦达的方法应用于几何问题。138然而，就符号而言，欧洲大陆的关键人物是勒内·笛卡尔，他在 1637 年出版的《几何学》一书中引入了一种与我们今天使用的符号非常相似的符号。笛卡尔推广了使用x、y和z表示未知数的惯例以及指数符号x ³。除了比哈里奥特的aaa更紧凑之外，这种符号最终还开辟了除整数之外的其他指数的可能性，特别是，它提供了一种统一指数和根的方法：。笛卡尔声称他完全没有受到韦达的影响，这一说法在学者中仍然存在争议。139无论如何，他为建立用于表达公式的完全“文字”符号做出了进一步的贡献。

Harriot and Pell were both located in England, where the mania for symbols was strongest in the early seventeenth century. Across the English Channel, Pierre de Fermat, Marino Ghetaldi, and Jean-Louis Vaulezard applied Viète’s methods to geometric problems.138 In regard to symbols, however, the crucial Continental figure was René Descartes, whose 1637 book Geometry introduced a notation recognizably like the one we use today. It was Descartes who popularized the convention of using x, y, and z for unknowns as well as the exponent notation x³. Apart from being more compact than Harriot’s aaa, this notation eventually opened the possibility of exponents other than integers and, in particular, of a way of unifying exponents and roots: . Descartes claimed not to have been influenced by Viète at all, a statement that remains controversial among scholars.139 Whatever the case may be, he contributed further to the establishment of a fully “literal” notation for expressing formulae.

新符号的一个实际问题是现代算法的重要先驱：公式。韦达的“种类”使得至少某些类型的计算过程能够完全通过符号来解释。例如，以迈克尔·达里的《利息缩影：复利与单利》（1677）为例。该书以简洁的形式呈现了复利计算的规则（图1.6）。图中所示的程序给出了基于以下因素的贷款本金：复利累积后的欠款金额，用一个抽象公式（a = ）表示，可以很快应用。这种计算规则的汇编与罗伯特·胡克等实验主义者的培根态度相一致，达里与胡克是朋友。140培根认为，科学对大众来说是难以理解的，但仍然会产生对他们有用的新技术。141符号公式非常适合这个模型：数学天才可以开发理论，这些理论将以实际操作的形式传递给普通人，普通人可以使用这些理论，而不必费心去想这一切是如何运作的。142

One practical issue of the new notation was an important precursor to the modern algorithm: the formula. Viète’s “species” made it possible to explain at least some types of computational procedure entirely through symbols. As an example, take Michael Dary’s Interest Epitomized, Both Compound and Simple (1677). This text presents rules for computations regarding compound interest in a compact form (figure 1.6). The procedure shown in the figure, which gives the principal of a loan based on the amount owed after the accrual of compound interest, is expressed in an abstract formula—(a = )—that one can apply in short order. The compilation of such computational rules aligned with the Baconian attitudes of experimentalists like Robert Hooke, with whom Dary was friendly.140 Bacon thought that science would be incomprehensible to the masses but would nonetheless produce new techniques that would be useful to them.141 Symbolic formulae fit this model nicely: mathematical geniuses could develop theories that would trickle down to commoners in the form of practical operations that they could use without having to trouble their heads about how it all worked.142

关于代数符号出现的记载，强烈倾向于辉格党式的叙事，这种叙事假定符号方法具有优越性。卡茨和帕夏尔推测，如果阿拉伯代数学家能够发展出符号，他们或许就能超越欧洲的同行。143但符号方法的优越性并非不证自明。被韦达贬低为“野蛮人”的伊斯兰数学家确实接触到了丢番图的著作——事实上，他们接触到的文献比欧洲人直到20世纪70年代才能接触到的还要多——但他们却很少关注这些符号。144 也许他们说得对。韦达的抽象方案——一次解决数百个问题，而不是一个问题——为追求“算法”的全面野心开辟了道路，如今，“算法”一词听起来越来越不祥。然而，这些野心最终会以这样的方式发展，并非命中注定。在追求韦达承诺的通用方法的过程中，17 世纪的数学家们绘制了两条截然相反的道路，而这两条道路都与我们所知的算法思维并不完全一致。

Accounts of the emergence of algebraic notation have shown a strong tendency toward Whiggish narratives that presume the superiority of symbolic methods. If only Arabic algebraists had developed symbols, Katz and Parshall speculate, they could have surpassed their European counterparts.143 But the superiority of symbolic methods is not self-evident. The Islamic mathematicians Viète denigrated as “barbarians” did gain access to Diophantus’s work—indeed, they had more of it than Europeans would see until the 1970s—and yet they paid little attention to the symbols.144 Perhaps they were right in that. Viète’s program of abstraction—solving problems by the hundred rather than one at a time—opened the way for the totalizing ambitions that now give the word algorithmic an increasingly ominous ring. That these ambitions should develop in the way they ultimately did, however, was not predestined. In pursuing the universal method Viète had promised, seventeenth-century mathematicians plotted out two divergent paths, neither of which quite aligns with algorithmic thinking as we know it.

走出文字的隐秘

Out of the Covert of Words

新的符号是否让代数更容易理解？这取决于人们所说的“理解”是什么意思。支持者强调符号表示法的紧凑性，正如罗伯特·胡克 (Robert Hooke) 所写，“它对推理和探究有巨大的用处，对理解和记忆也有很大的帮助。” 145这些认知优势并不局限于数学领域：胡克希望自然历史也能用类似的简写来表达，“用尽可能少的字母或字符来表达大量的情况。” 146威廉·奥特雷德 (William Oughtred) 是一位颇具影响力的数学家，他在英国推广了似是而非的算术，他走得更远。在1647年出版的《新造数学之钥》（他早期拉丁语教科书的英译本修订版）中，他提出符号可以揭示语言面纱之下的赤裸裸的真理：“为了更清晰地观察事物本身，我将命题和论证从文字的掩蔽中解放出来，用直观的注释和种类来表达它们。” 147奥特雷德声称自己是从文字中“解放”思想，将符号定位为一种中性的、没有修辞的媒介——赤裸裸的真理，不加修饰，只有实质，没有风格。如果真是这样，它们似乎是对培根式语言抱怨的完美补救。

Did the new symbols make algebra easier to understand? It depends on what one means by understand. Advocates emphasized the compactness of symbolic notation, which, as Robert Hooke wrote, “is of huge Use in the Prosecution of Ratiocination and Inquiry, and is of vast Help to the Understanding and Memory.”145 These cognitive advantages were not specific to mathematical fields: Hooke wishes that natural history could be expressed in a similar shorthand employing “as few Letters or Characters as it has considerable Circumstances.”146 William Oughtred, an influential mathematical practitioner who popularized specious arithmetic in England, went further. In the 1647 book The Key of the Mathematicks New Forged and Filed (a revised English translation of his earlier Latin textbook), he suggests that symbols can reveal the naked truth beneath the veil of language: “Wherefore that I might more cleerly behold the things themselves, I uncasing the Propositions and Demonstrations out of their covert of words, designed them in notes and species appearing to the very eye.”147 By claiming to be “uncasing” ideas from words, Oughtred positions the symbols as a neutral medium devoid of rhetoric—the naked truth, unadorned, all substance, no style. If so, they would seem a perfect remedy to the Baconian complaint about language.

然而，这些符号并不总是清晰易懂。奥特雷德的学生之一，牛津天文学家塞思·沃德，在1654年的一本关于三角学的书中指出了符号的潜在晦涩：“我很清楚，这本小书大部分内容都避免使用文字，力求将事物本身直接传达给读者理解，有些人会认为它晦涩难懂：事实上，它主要是为了那些已经熟悉这类名称的人（前提是他们没有疏忽）而编写的。” 148这段话展现了一种张力，这种张力在几个世纪以来的符号方法讨论中一直持续存在，直到编程语言的出现。这些符号似乎能够精确地表达数学思想。和透明度，但对于未经训练的读者来说，它们可能完全不透明。149换句话说，能够很好地满足认知功能的符号并不一定同样适用于交流；对作者来说很清楚的铭文，对其他人来说可能毫无意义。

Yet the symbols did not always appear clear or comprehensible. One of Oughtred’s pupils, the Oxford astronomer Seth Ward, notes the potential opacity of symbols in a 1654 book on trigonometry: “It does not escape me that this little book, which abstains for the most part from words and strives to carry the things themselves to the understanding at once, will be seen by some as portentous and difficult: Truly it was produced chiefly for the sake of those to whom (provided they did not neglect themselves) designations of this kind have already become familiar.”148 This passage exhibits a tension that would continue to pervade discussions of symbolic methods for centuries, all the way to the emergence of the programming language. The symbols seemed to represent mathematical ideas with precision and transparency, yet to the untrained reader they could be altogether opaque.149 To put it another way, notations that serve cognitive functions well do not necessarily work equally well for communication; inscriptions that seem clear to their writers might make no sense to others.

这一时期的主要数学家大多强调符号的个人使用，很少考虑交流问题。在 1637 年的《几何学》中，笛卡尔展示了如何将几何问题转化为代数问题，反之亦然；正是在这里，他介绍了他版本的笛卡尔坐标系，尽管他的系统与我们现在所知的系统有很大不同。150这项工作被认为是他在《方法论》中描述的推理方法的一个例子，在该方法中，他决心“在我的判断中，只包含那些清晰明确地呈现在我脑海中以排除一切疑问的东西。” 151正如马修·琼斯所强调的那样，笛卡尔所处的知识文化贬低被视为机械的活动，包括代数计算。152因此，笛卡尔将符号视为一种使问题更容易在脑海中理解的方式，而不是一种产生结果的工具手段。

The major mathematicians in the period largely emphasized individualistic uses of symbols and gave little thought to matters of communication. In his 1637 Geometry, Descartes shows how geometric problems may be translated into algebraic ones and vice versa; it is here that he introduced his version of what is now known as the Cartesian coordinate system, although his system was rather different from the one we now know.150 This work was supposed to be an example of the method of reasoning that he describes in Discourse on Method, in which he resolves “to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.”151 As Matthew L. Jones has emphasized, Descartes was situated in an intellectual culture that devalued activities perceived as mechanical, including algebraic calculation.152 Descartes accordingly embraced symbols less as an instrumental means of producing results than as a way of making problems easier to grasp mentally.

笛卡尔在其早期著作《我们天生智力的指导规则》 （约 1628 年）中详细讨论了符号。这部著作直到 1701 年才出版，尽管莱布尼茨在 1670 年获得了它的手稿。153笛卡尔写道，在所有科学中，“只有算术和几何不受任何虚假或不确定性的影响”，因为它们完全通过演绎而非经验获得知识；因此，他将它们视为一种他称之为“通用数学”的方法的典范。154与韦达一样，笛卡尔将代数视为古人用来解决问题的一种秘密艺术。155笛卡尔认为，这种艺术的力量可以超越丢番图试图解决的玩具问题，创造出“一种比人类留给我们的任何其他知识工具都更强大的知识工具”。156这里出现了我们在韦达和佩尔的著作中已经看到的普遍化野心的另一个例子：认为一种“分析”方法可以用来解决所有问题。157

Descartes discusses symbols at length in his early text Rules for the Direction of Our Native Intelligence (c. 1628). This work was not published until 1701, although Leibniz acquired a manuscript copy of it in 1670.153 Of all sciences, Descartes writes, “Arithmetic and Geometry alone are free from any taint of falsity or uncertainty” because they derive knowledge wholly through deduction, not from experience; he thus takes them as a model for a method he calls “universal mathesis.”154 Like Viète, Descartes identifies algebra with a secret art the ancients supposedly used in solving problems.155 Descartes thinks the power of this art could be extended beyond the toy problems Diophantus was out to solve, producing “a more powerful instrument of knowledge than any other that has been bequeathed to us by human agency.”156 Here appears another instance of the universalizing ambitions that we already saw in Viète and Pell: the idea that a single method of “analysis” could be used to solve all problems.157

作为笛卡尔通用数学的一部分，符号表示法之所以重要，主要是因为其紧凑的形式可以帮助人们将复杂的想法形象化。在《规则》中，笛卡尔强调了“在连续不间断的思维活动中”反复思考的重要性。158其中一种方法是使用“简明缩写”，省略对象的不相关方面。159例如，他引入了一种类似于几何的简单代数符号，其中小写字母代表已知量，大写字母代表未知量。160与当时的算法文本一样，他首先建立了符号和文字：他写道，表达式 2 a ^{3 “将等同于‘用字母}a表示的量的两倍，包含三个关系’这样的文字。” 161他解释说，符号优于文字的地方在于，它们提出问题的方式既不包含任何“多余的”内容，也不会“通过要求大脑同时掌握许多事物，而白白浪费我们的心智能力” 。162

As a part of Descartes’s universal mathesis, symbolic notation was primarily important because its compact form could help people visualize complex ideas. In Rules, Descartes emphasizes the importance of running ideas over “in a continuous and uninterrupted act of thought.”158 One way of doing so is to employ “compendious abbreviations” that leave out irrelevant aspects of objects.159 As an example, he introduces a simple algebraic notation somewhat like that of Geometry, in which lowercase letters represent known quantities and uppercase letters unknowns.160 As in the algorism texts of the time, he begins by establishing an equivalency between the symbols and words: the expression 2a³, he writes, “will be the equivalent of the words ‘the double of the magnitude which is symbolized by the letter a, and which contains three relations.’”161 The advantage of the symbols over these words, as he explains it, is that they present a problem in a way that contains nothing “superfluous” and nothing that would “exercise our mental powers to no purpose, by requiring the mind to grasp a number of things at the same time.”162

只有结合笛卡尔对身心二元论的坚持，才能充分理解对符号认知功能的这种解释。与克莱因的论述相反，笛卡尔并不将符号公式等同于数学知识本身。163在《规则》的开篇，笛卡尔指出，将“完全在于心灵认知锻炼的科学与依赖于身体锻炼和调动的艺术”相提并论是错误的。164这种二元论将哲学与当时教授的计算实践对立起来，在计算实践中，意义问题与物理笔法混为一谈。真正的科学是关于清晰而独特的感知，对笛卡尔而言，这些感知与任何特定的书写形式无关。在介绍他的符号表示法之前，他还提出了一些其他表示数量的方式，这些方式与波爱修斯的算术理论有更多共同之处，例如六个点排列成一个三角形；他还表明，算术运算可以用几何图形来思考。165他明确表示，每一个“数字”都是众多表现形式中的一种；数字本身就是思想的对象。166

This explanation of the cognitive function of symbols can only be fully understood in light of Descartes’s commitment to mind–body dualism. Contrary to Klein’s account, Descartes does not identify symbolic formulae with mathematical knowledge itself.163 In the opening paragraph of Rules, Descartes states that it is an error to compare “the sciences, which entirely consist in the cognitive exercise of the mind, with the arts, which depend on an exercise and disposition of the body.”164 This dualism places philosophy in opposition to the computational practices that were taught at the time, in which matters of meaning were mixed up with physical pen skills. True science was about clear and distinct perceptions, which were not, for Descartes, tied to any particular form of writing. Before presenting his symbolic notation, he also suggests some other ways of representing quantities that have more in common with Boethius’s theory of arithmetic, such as six points arranged in a triangle; he also shows that arithmetical operations can be thought of in terms of geometric diagrams.165 Each of these “figures,” he makes it clear, is one form of representation among many; the numbers themselves are objects of thought.166

笛卡尔对概念清晰度的关注将持续影响几个世纪以来关于数学符号的讨论，尤其是在法国。但他对符号功能的描述只是近代早期出现的众多观点之一。另一种观点出现在他不太出名的同时代人皮埃尔·埃里戈内 (Pierre Hérigone) 的著作中。埃里戈内唯一的主要著作是六卷本的《数学全书》 ( Cvrsvs mathematicvs , 1634–42)，该书旨在概述拉丁语和法语中的所有数学。在书的开头，他介绍了代数和几何的缩写和符号，其中一些今天我们熟悉，而另一些则不常见 (图 1.7 )。这种广泛的符号使埃里戈内成为符号代数批评者的目标，其中包括艾萨克·巴罗 (Isaac Barrow)；20 世纪历史学家弗洛里安·卡乔里 (Florian Cajori) 认为埃里戈内“几乎不顾一切地急于引入一套详尽的符号”。167他声称，他开发这些符号的目标是使数学“简洁易懂，无需使用任何语言” 。168

Descartes’s focus on conceptual clarity would continue to influence discussions of mathematical symbols for centuries, especially in the French context. But his account of the function of symbols was only one of a number of views that arose in the early modern period. An alternative appears in the work of his lesser-known contemporary Pierre Hérigone. Hérigone’s only major work is the six-volume Cvrsvs mathematicvs (1634–42), which aims to give an overview of all of mathematics in both Latin and French. At the beginning of the book, he introduces abbreviations and symbols for both algebra and geometry, some of which are familiar today and some of which are not (figure 1.7). This extensive notation made Hérigone a particular target for critics of symbolic algebra, including Isaac Barrow; the twentieth-century historian Florian Cajori attributed to Hérigone “an almost reckless eagerness to introduce an exhaustive set of symbols.”167 His goal in developing these symbols, as he states, is to make mathematics “brief and intelligible, without the use of any language.”168

埃里戈内的作品对符号的功能提出了比笛卡尔更广阔的视野。169笛卡尔将他的符号置于段落文本；甚至连符号迷 Harriot 也使用拉丁词如sit和ergo将方程式拼凑成一个论证。Hérigone 发明了符号和缩写来取代数学证明中的这些文字元素，使得（至少在原则上）整个证明可以在没有任何文字的情况下得到开发和交流。在标题页上，Hérigone 声明他的目的是以“真实和通用的注释”来呈现数学。170从文本的多语言性质可以看出他所说的“通用”。在介绍符号时，他用拉丁语和法语解释它们：~ 是“减号，moins ” 。171甚至他的缩写设计得既像拉丁语单词又像法语单词（尽管并不完美）。“Pr.”可以读作primus或premier；“reg.”可以读作regula或reigle。172符号的跨语言性质对于本书的结构至关重要。他的散文解释以拉丁语和法语出现，但在他的符号证明中，他通常提供一个单一版本，拉丁语和法语读者都可以阅读。他试图创造一个真正的角色——为数学配备一套符号，像汉字一样，可以被多种语言的使用者阅读。

Hérigone’s work presents a more expansive vision for what symbols could do than that of Descartes.169 Descartes situated his notations within paragraphs of text; even the symbol-besotted Harriot used Latin words like sit and ergo to stitch equations together into an argument. Hérigone invented symbols and abbreviations to replace these verbal elements of mathematical proofs, enabling (at least in principle) an entire proof to be developed and communicated without any words at all. On the title page, Hérigone states that his purpose of presenting mathematics in “notes real & universal.”170 What he means by “universal” is apparent from the polyglot nature of the text. When introducing the symbols, he explains them in both Latin and French: ~ is “minus, moins.”171 Even his abbreviations are designed so as to resemble (albeit imperfectly) both Latin and French words. “Pr.” may be read as either primus or premier; “reg.” is either regula or reigle.172 The translingual nature of the symbols is crucial to the structure of the book. His prose explanations appear in both Latin and French, but in his symbolic proofs, he usually presents a single version that is supposed to be readable by both Latinate and Francophone audiences. He was trying to create a real character—to outfit mathematics with a set of symbols that, like kanji, could be read by speakers of multiple languages.

埃里戈内的作品是一种趋势的早期表达，这种趋势从 17 世纪后期的莱布尼茨到 20 世纪的查尔斯·K·布利斯一再重复出现：希望通过用符号代替文字来改善交流。173然而，17世纪中叶埃里戈内等人实现这一目标的方式存在一个问题，这个问题最终导致了哲学上的反思。尽管埃里戈内的符号应该能够实现“普遍”交流，但它们仍然是由文字定义的。因此，它们的普遍性取决于拉丁语和法语是否存在共同的思想体系。如果这两种语言无法协调——比如说，拉丁语tangit （它接触、它到达、它影响）与法语elle touche （她/它接触）的意思不完全相同——那么清晰的跨语言交流实际上并没有建立。174十七世纪的数学家们大多确信，他们所处理的概念是普遍可理解的，因此解释符号的含义并非难事。然而，有迹象表明，沟通比真实字符观点所揭示的要困难得多。

Hérigone’s work was an early expression of a tendency that would recur time and again, from Leibniz in the later seventeenth century to Charles K. Bliss in the twentieth: a desire to improve communication by replacing words with symbols.173 There was, however, a problem with the way Hérigone and others in the mid-seventeenth century approached this goal that would ultimately lead to a philosophical reckoning. Even though Hérigone’s notations are supposed to enable “universal” communication, they are still defined by words. Their universality thus depends on the existence of a common stock of ideas shared by Latin and French. If the two languages fail to align—if, say, the Latin tangit (it touches, it reaches, it affects) does not suggest quite the same thing as the French elle touche (she/it touches)—then clear translingual communication has not actually been established.174 Seventeenth-century mathematicians were mostly confident that the concepts they dealt with were universally intelligible and that, as a result, explaining the meanings of symbols would not be a significant problem. Yet hints were bubbling up that communication was harder than the real-character view let on.

最早提出此类观点的论点之一来自托马斯·霍布斯，他因对几何学持有奇怪的观点而与牛津大学数学家约翰·沃利斯结下了不解之缘。175霍布斯在其 1656 年出版的《给数学教授的六堂课》一书中从多方面批评了沃利斯的数学著作，其中之一就是他对符号的过度依赖。霍布斯写道，符号在解开证明的细节时可能有用，但它们“不应该出现在公共场合，就像你在办公室里做的最不体面的必要事情一样” 。176这种粗俗的说法意味着，用霍布斯在其他地方的说法，符号仅仅是“标记”，而不是真正的“符号”——它们可以帮助选择它们的人的记忆，但缺乏向他人传达意义所需的社会依据。177霍布斯后来将代数符号称为“一种非常狭隘的语言”，暗示符号与文字的区别并不像真实字符的倡导者所声称的那样大。178对霍布斯来说，符号可以规避语言需求的想法是不合理的；所有代数学家都在采用符号正在创造一种只有他们的独家小圈子才能理解的秘密。

One of the earliest arguments along these lines came from Thomas Hobbes, whose eccentric views about geometry embroiled him in an extended feud with the Oxford mathematician John Wallis.175 In his 1656 book Six Lessons to the Professors of the Mathematiques, Hobbes criticizes Wallis’s mathematical work along a number of lines, among them his overreliance on symbols. Symbols, Hobbes wrote, might be useful in working out the details of a proof, but they “ought no more to appear in publique, then the most deformed necessary business which you do in your Chambers.”176 This scatological statement implies that the symbols are, in terms Hobbes laid out elsewhere, mere “marks” rather than true “signs”—they can aid the memory of the person who chose them, but they lack the social warrant needed to convey meanings to others.177 Hobbes later refers to algebraic notation as “a very narrow Language,” suggesting that symbols were not as different from words as the advocates of real characters claimed.178 For Hobbes, the idea that symbols could circumvent the need for language was illegitimate; all the algebraists were doing by adopting symbols was creating a secret cant comprehensible only to their exclusive clique.

在计算史前史的叙述中，霍布斯的反符号论战通常被忽视。人们更多地关注他 1655 年出版的《哲学要素》一书，他在书中进行了前所未有的比较，似乎预见了认知科学的后期发展：“我所说的RATIOCINATION是指计算。” 179霍布斯认为，所有推理都包含以类似于加减的方式组合和分裂想法；因此，他将该书的第一部分命名为“计算或逻辑”。180这种引人注目的比较使得哲学家 Hubert Dreyfus 和 John Haugeland 都将霍布斯视为人工智能的奠基人。181霍布斯可能确实引导他的一些早期读者走上了这条道路；他的言论可能是莱布尼茨逻辑演算的灵感来源。但正如霍布斯对符号代数的批判所表明的那样，将思维视为符号操作机器并不意味着创建一个外化的符号推理系统是个好主意，无论它是纸质的还是金属的。如果“人工智能”一词对霍布斯有任何意义的话，它应该指的是国家——由众多人类共同协作而产生的复杂集合体。从霍布斯的角度来看，要使其结果具有权威性，符号操作机器必须以通用语言为基础，而代数符号（至少在他看来）并非如此。

Hobbes’s antisymbolic polemics have generally been overlooked in accounts of the prehistory of computation. Much more has been made of his 1655 book Elements of Philosophy, in which he makes an unprecedented comparison that seems to anticipate much later developments in cognitive science: “By RATIOCINATION, I mean Computation.”179 According to Hobbes, all reasoning consisted of combining and splitting ideas in a manner analogous to addition and subtraction; he thus titles the first section of the book “Computation or Logique.”180 This striking comparison has led the philosophers Hubert Dreyfus and John Haugeland each to frame Hobbes as a founding figure of artificial intelligence.181 Hobbes may indeed have led some of his early readers down this track; his statements were probably an inspiration for Leibniz’s logical calculus. But as Hobbes’s attack on symbolic algebra shows, viewing the mind as a symbol-manipulating machine does not entail that it would be a good idea to create an externalized system of symbolic reasoning, be it made of paper or of metal. If the phrase “artificial intelligence” would have meant anything to Hobbes, it would have meant the state—the complex aggregate created when many human beings work in consort. For its results to be authoritative from a Hobbesian perspective, a symbol-manipulating machine would have to be grounded in a common language, which algebraic symbols (at least in his opinion) were not.

霍布斯的论战未能遏制符号的浪潮，至少在初期如此。教科书自信地认为，符号的“意义”将轻易地为所有人所理解。就数字而言，这种自信或许是有道理的；就“7”的含义达成共识，一般来说并非什么大问题。虽然代数带来了更大的解释难度，但符号相对于文字的优势（尽管霍布斯持不同意见）难以否认，而且将这些优势扩展到其他领域的愿望也十分普遍。然而，一个严肃的问题是，符号与文字的独立性不如奥特雷德和沃德的修辞所暗示的那样。沃德在其三角学著作的开篇，将他的符号映射到拉丁语单词上，例如，将R解释为“半径” 。182因此，在声明他的著作“放弃”使用文字，而倾向于使用直接表达事物的符号之后，他将这些符号定义为——文字。使用文字来定义符号在很大程度上没有受到真实字符倡导者的注意，但它为将符号符号从一种单纯的专业书写形式发展为一种成熟的语言替代品的尝试带来了问题。

Hobbes’s polemics did little to stem the tide of symbols, at least at first. Textbooks confidently assumed that the “significations” of symbols would readily become transparent to all. As far as numerals went, this confidence may have been justified; establishing an agreement about what 7 meant was not, generally speaking, a significant problem. While algebra raised larger interpretive difficulties, the symbols had advantages over words that were (Hobbes’s dissent notwithstanding) hard to deny, and the desire to extend these advantages to other areas was widespread. Yet there was a serious case to be made that symbols were less autonomous from words than the rhetoric of Oughtred and Ward let on. At the beginning of his trigonometry book, Ward maps his symbols onto Latin words, explaining R, for instance, as “Radius.”182 Thus, just after stating that his book “abstains” from the use of words in favor of symbols that express things directly, he defines those symbols with—words. The use of words in defining symbols went, for the most part, unnoted by advocates of real characters, but it posed a problem for attempts to develop symbolic notation from a mere specialized form of writing into a full-fledged alternative to language.

这种紧张关系在莱布尼茨的著作中达到了顶峰。就像这些提议一样莱布尼茨的普遍特征被认为可以为从神学到法律等各个领域的推理提供符号方法。这一特征只是莱布尼茨探索符号所开启的可能性的众多项目之一。然而，自 16 世纪以来日益增长的理论担忧在莱布尼茨之后更是雪上加霜，他的追随者们努力为他的微积分版本寻找基本原理，而牛顿和其他人则攻击他使用“机械”方法。莱布尼茨的工作将 17 世纪关于符号的思想推向了极致，标志着对“字符”力量有些天真的信念的终结，以及一个漫长历史进程的开始，在这个进程中，经过几个世纪的时间，算法思维的界限将不断被争论并最终被确定。

This tension reached its crisis in the work of Leibniz. Like the proposals of Pell and Hooke, Leibniz’s universal characteristic was supposed to provide symbolic methods for reasoning about subjects ranging from theology to law. The characteristic was only one of a plethora of projects in which Leibniz explored the possibilities opened by symbols. Yet the theoretical concerns that had been mounting since the sixteenth century only worsened in Leibniz’s wake, as his followers struggled to find a rationale for his version of calculus, and as Newton and others attacked him for employing “mechanical” methods. In pushing seventeenth-century ideas about symbols to their utmost limit, Leibniz’s work marks the end of the somewhat naïve belief in the power of “characters” and the beginning of the long historical process by which, over the course of centuries, the boundaries of algorithmic thinking would be contested and ultimately set.

梦想

The Dream

思考戈特弗里德·威廉·莱布尼茨在计算史前史中的地位，如同拨开一层时代错位的浓雾。莱布尼茨设计了早期的机械计算器，后来的评论家们在他的著作中发现了许多后来对计算机科学至关重要的思想，从布尔逻辑到二进制数字。20世纪中叶，诺伯特·维纳认为莱布尼茨可能是“控制论的守护神”，从而确立了莱布尼茨在计算机史经典中的地位。控制论是研究人机控制与交流系统的领域。1莱布尼茨自此成为计算史学界的常客：著名作家马丁·戴维斯将通用计算机的概念称为“莱布尼茨的梦想”，而沃尔夫冈·托马斯在其算法史论述中，则探讨了算法范畴的近期扩展是否意味着我们正在“回归莱布尼茨的愿景”。2虽然莱布尼茨的思想确实与计算机科学的关注点有重叠，但它也反映出一种与现代观念截然不同的世界观，而这一事实在人们试图将莱布尼茨塑造成一位奠基人时往往会被忽视。

To consider Gottfried Wilhelm Leibniz’s position in the prehistory of computation is to peer through a thick fog of anachronism. Leibniz designed an early mechanical calculator, and later commentators have found in his work anticipations of numerous ideas that would later become important to computer science, from Boolean logic to binary numerals. In the mid-twentieth century, Norbert Wiener ensured Leibniz’s place in the computer history canon by suggesting that he could be “a patron saint for cybernetics,” the field that deals with systems of control and communication involving both humans and machines.1 Leibniz has since become a fixture in histories of computation: the popular writer Martin Davis refers to the idea of a general-purpose computer as “Leibniz’s dream,” and in his account of the history of algorithms, Wolfgang Thomas considers whether the recent expansion of the category of algorithm means we are “Returning to Leibnizian Visions.”2 While Leibniz’s thinking does overlap with the concerns of computer science, it also reflects a worldview quite alien to modern sensibilities, and this fact tends to get lost amid attempts to make Leibniz into a founding figure.

莱布尼茨设想过类似现代计算机的东西，这种观点很大程度上源于他的论文《普通科学序言》（1677 年）和《发现的艺术》（1685 年），以及类似的评论。在其他一些地方。3在这两篇文章中，莱布尼茨表达了一个梦想，正如我在第一章中所展示的，这个梦想在17世纪很普遍：将符号的力量扩展到新的知识领域。莱布尼茨设想，这个项目将由一组被称为“普遍特征”的符号以及一个推理演算器（一种理性演算）组成，任何问题都可以用它来解决。在1685年的文章中，莱布尼茨解释道：

The belief that Leibniz imagined something like a modern computer stems in large part from his essays “Preface to the General Science” (1677) and “The Art of Discovery” (1685), along with similar remarks in a few other places.3 In both texts, Leibniz expresses a dream that was, as I showed in chapter 1, common in the seventeenth century: extending the power of symbols to new areas of knowledge. This project, as Leibniz conceives it, would consist of a set of symbols called a universal characteristic along with a calculus ratiocinator—a rational calculus—by which any question can be resolved. In the 1685 essay, Leibniz explains:

纠正我们推理的唯一方法是使它们像数学家的推理一样具体，以便我们一眼就能发现自己的错误，而当人们之间发生争论时，我们可以简单地说：让我们不用多说，计算一下，看看谁是对的。4

The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, in order to see who is right.4

这段话经常被断章取义，暗示“让我们计算”（calculemus）是指类似可编程计算机的东西。但目前尚不清楚他是否将这种“计算”理解为现代意义上的算法。正如马修·琼斯 (Matthew L. Jones) 所表明的，莱布尼茨的一些努力至少涉及对表格数据中的模式进行归纳推理。5此外，莱布尼茨对这个系统的目标在某些方面比现代计算机的目标更为雄心勃勃。莱布尼茨在《发现的艺术》中描述的系统不仅仅是给出答案；它还应该以一种让人无法怀疑的方式展示它们。这个想法与 21 世纪的黑箱算法相去甚远。莱布尼茨的计算机（如果这种事真的可能实现的话）只会计算真相。

This passage is often quoted out of context with the implication that the “Let us calculate” (calculemus) refers to something like a programmable computer. But it is not clear that he understood this “calculation” to be algorithmic in the modern sense. At least some of Leibniz’s efforts, as Matthew L. Jones has shown, involved inductive reasoning about patterns in tabular data.5 Moreover, Leibniz’s goals for this system were in some regards more ambitious than those of modern computing machines. The system Leibniz describes in “The Art of Discovery” is not just supposed to spit out answers; it is supposed to demonstrate them in a way that makes them impossible to doubt. This idea is worlds away from the black box algorithms of the twenty-first century. A Leibnizian computing machine, if such a thing were really possible, would only compute the truth.

本章从历史背景出发，阐述了莱布尼茨在算法思维发展中所扮演的角色。本章特别关注两个涉及符号的项目：普遍特征及其在数学符号方面的工作。在第一章中，我指出，17 世纪的数学家将数字和代数符号描述为与字母或汉字相当的书写系统。将这些符号视为文字而非语言，突显了它们可以用多种方言朗读，同时（表面上）对每个人都具有相同的含义；这使得它们在当时被称为真正的字符。虽然莱布尼茨并不打算将普遍特征视为严格意义上的真正字符，但他的作品展现了类似的思维方式：文字与特定社群的历史密不可分，因此不可避免地会产生分歧，而莱布尼茨的符号则应该锁定在每个人都能认同的思想上。莱布尼茨用这种普遍性来解释符号方法在微积分等数学领域的认知能力。但对这些方法在莱布尼茨数学著作的早期接受中浮现出来，具有讽刺意味的是，这导致了激发他乌托邦计划的符号信仰的崩溃。

This chapter gives a historically contextual account of Leibniz’s role in the development of algorithmic thinking. It focuses, in particular, on two projects involving symbols: the universal characteristic and his work on mathematical notation. In chapter 1, I showed that seventeenth-century mathematicians described numerals and algebraic symbols as writing systems comparable to the alphabet or to the Chinese han characters. Viewing these symbols as writing rather than language highlighted the fact that they could be read aloud in multiple vernaculars while bearing (apparently) the same meanings for everyone; this made them, in period terms, real characters. While Leibniz did not intend the universal characteristic to be a real character in the strict sense, his work exhibits a similar way of thinking: whereas words were inextricable from the histories of particular communities and thus inevitably divisive, Leibniz’s symbols were supposed to latch onto ideas about which everyone could agree. Leibniz used this universalism to explain the cognitive power of symbolic methods in mathematical fields such as calculus. But doubts about the conceptual foundations of these methods surfaced in the early reception of Leibniz’s mathematical work, which contributed, ironically, to a collapse of the faith in symbols that had inspired his utopian project.

只有结合莱布尼茨的哲学和政治立场，才能充分理解他的符号观。在莱布尼茨的形而上学中，他把人类灵魂称为单子，它们并不直接互动，而是存在于一种预先设定的相互之间以及与物质世界的和谐之中。6这种对和谐的强调在莱布尼茨的语境中具有政治意义。玛丽亚·罗莎·安托尼亚扎在她所写的莱布尼茨传记中强调了莱布尼茨的成长经历对其哲学的影响。7莱布尼茨出生于萨克森州，当时属于神圣罗马帝国的一部分。与法国的中央集权君主制相反，帝国由自治的、有时甚至是对立的公国组成，这些公国有着不同的文化认同和宗教信仰。在政治和宗教方面，莱布尼茨强烈倾向于和解——在对立的派别和教派之间寻找共同点。安托尼亚扎毕生致力于“共融主义”，即致力于使新教徒和天主教徒重新团结起来。他将“普世性”视为这一理念不可或缺的一部分，并将其视为统一知识体系的基础，从而将神学建基于不容置疑的原则，弥合宗教分歧。

Leibniz’s views about symbols can only be fully understood in the context of his philosophical and political commitments. In Leibnizian metaphysics, human souls, which he called monads, have no direct interactions with other beings but exist in a preestablished harmony with one another and with the material world.6 This emphasis on harmony had a political meaning in Leibniz’s context. In her biography of Leibniz, Maria Rosa Antognazza has emphasized the influence of Leibniz’s upbringing on his philosophy.7 Leibniz was born in Saxony, which was then part of the Holy Roman Empire. In contrast to the centralized monarchy of France, the empire consisted of autonomous and sometimes antagonistic principalities with varying cultural identities and religious affiliations. Politically and religiously, Leibniz tended strongly toward reconciliation—toward finding common ground between antagonistic factions and sects. Antognazza has demonstrated his lifelong commitment to irenicism, meaning the project of reuniting the Protestants and the Catholics. He viewed the universal characteristic as an integral part of this project, a basis for a unified body of knowledge that would bridge religious divides by grounding theology in indisputable principles.

他对语言的态度也类似。17 世纪后期，英语和法语正走向标准化，因为词典和语法将大都市（伦敦和巴黎）和大学的方言编纂成默认方言。8当时，神圣罗马帝国大部分地区都使用德语，但标准化程度较低，而且帝国是一个多语言国家，包括使用意大利语、捷克语、意第绪语和其他语言的群体。9跨地区交流用书面语比用口语更容易，因为德语的书面形式并不能反映方言之间的所有差异；事实上，18 世纪形成的标准德语最初完全是书面语。10这一特点或许可以被视为一种更适合当时语境的标准化替代方案——并非试图将一种方言确立为通用语，而是试图在所有语言和方言之间业已存在的和谐基础上建立一种新的交流模式。

Something similar may be said of his attitude toward language. English and French were moving toward standardization in the late seventeenth century, as dictionaries and grammars codified the dialects of the metropoles (London and Paris) and of the universities as defaults.8 The German language, which was spoken across much of the Holy Roman Empire, was less standardized at the time, and the empire was multilingual, also including communities speaking Italian, Czech, Yiddish, and other languages.9 Communication across regions would have been easier in writing than in speech, since written forms of German did not reflect all the differences between dialects; indeed, the standard German that would form in the eighteenth century was at first exclusively written.10 The characteristic might be seen as an alternative to standardization that is more suited to this context—an attempt not to instate one dialect as the lingua franca but rather to base a new mode of communication on the harmony that already existed among all languages and dialects.

尽管这一普遍性最终声名鹊起，但它只不过是一些零散的笔记和暗示性的评论。用现代的话来说，它不过是些空谈。就其影响力而言，莱布尼茨对计算史最重要的贡献是他在无穷小微积分方面的工作，这极大地扩展了符号的功能范围。1684 年，莱布尼茨在题为《求最大值和最小值的新方法》的文章中介绍了他的微积分版本，并将该方法称为算法。11莱布尼茨的数学创新在随后的几十年中产生了两种相反的效果。莱布尼茨称之为“算法”的技术可以做出令人惊叹的事情，解决了许多困扰古人的问题；因此，他的工作加剧了人们（尤其是在欧洲大陆）对符号的兴趣，这种兴趣在 17 世纪一直不断增加。但莱布尼茨并没有（至少在公开场合）令人满意地解释他的方法为何有效，因此符号方法受到了更大程度的批判性审查，在某些圈子里甚至遭到怀疑。

Famous as it eventually became, the universal characteristic amounted to little more than scattered notes and suggestive comments. It was, in modern terms, vaporware. In terms of influence, Leibniz’s most important contribution to the history of computation was his work on the infinitesimal calculus, which greatly expanded the scope of what symbols could do. The 1684 article in which Leibniz introduced his version of calculus, titled “New Method for Maximums and Minimums,” called this method an algorithm.11 Leibniz’s mathematical innovations had two countervailing effects in the ensuing decades. The technique that Leibniz called an “algorithm” could do astonishing things, solving numerous problems that had puzzled the ancients; his work thus heightened, especially in Continental Europe, the excitement about symbols that had been building over the course of the seventeenth century. But Leibniz did not articulate (at least publicly) a satisfactory explanation of why his method worked, and so symbolic methods came to be subject to a greater degree of critical scrutiny and, in some circles, skepticism.

莱布尼茨在一系列公开辩论中详细讨论了这些问题。12他在微积分方面的工作使他与艾萨克·牛顿展开了长时间的交锋，因为他们都声称自己是这一思想的先驱。尽管牛顿也使用了符号，但他比莱布尼茨更怀疑符号方法的价值；据说，牛顿曾说，即使是“笨手笨脚的人”，也能运用代数的机械方法。13莱布尼茨也曾在与约翰·洛克的争论中为自己对符号的使用进行辩护。洛克在其《人类理解论》（1689）中提出了一种原始人类学相对主义，这削弱了莱布尼茨关于符号能够清晰地传达思想的主张。争论的焦点恰恰在于普遍性特征的可能性：洛克的立场意味着，语义约定的必要性是不可避免的，仅仅使用符号形式不足以确保人们真正思考的是相同的事情。总体而言，洛克一方获胜了。到了十八世纪，莱布尼茨终其一生都坚持认为符号能够表达前语言的思想，但这一观点已不再被广泛接受。然而，莱布尼茨的方法依然存在——他的微积分“算法”显然行之有效，尽管人们对其概念基础众说纷纭。因此，一个多世纪以来，“算法”的地位始终取决于一个关于语言的问题：如何才能使符号有意义？

Leibniz hashed out these issues in a series of public debates.12 His work on calculus embroiled him in an extended interchange with Isaac Newton, since both claimed credit for developing the idea first. Although he did employ notation, Newton was more skeptical of the value of symbolic methods than Leibniz; even a “bungler,” he reportedly said, can use the mechanical methods of algebra.13 Leibniz also had occasion to defend his use of symbols in a dispute with John Locke. In his Essay Concerning Human Understanding (1689), Locke had advanced a kind of protoanthropological relativism that undermined Leibniz’s claims about the power of symbols to convey ideas transparently. At issue was nothing less than the possibility of the universal characteristic: Locke’s position entailed that there was no getting around the need for semantic convention, that mere symbolic forms were not enough to ensure that people were genuinely thinking the same things. Broadly speaking, Locke’s side won. In the eighteenth century, it would no longer be widely accepted that symbols could express prelinguistic ideas, as Leibniz maintained to the end of his life. Yet Leibniz’s methods remained—the “algorithm” of his calculus apparently worked, even though people could not agree on its conceptual basis. For over a century, then, the status of “algorithms” hinged on a question about language: what does it take to make symbols meaningful?

反对语言的设计

A Design against Language

正如我在第一章中提到的，包括约翰·佩尔和罗伯特·胡克在内的一些十七世纪思想家，设想了一种能够用代数符号清晰地表达任何事物的系统。另一个类似的提议（尽管指的是另一种符号）来自勒内·笛卡尔。在1629年写给朋友马林·梅森的一封信中，笛卡尔指出，印地语-阿拉伯数字仅用十个符号组合起来就能表达无限多的数字。按照简单的规则。“在一天之内，”他写道，“人们就能学会命名无限系列数字中的每一个，从而用一种未知的语言写出无限多的单词。对于表达人类思维范围内所有其他事物所需的所有其他词汇，我们也能做到同样的事情。” 14莱布尼茨的普遍特征至少在某些版本中与这一提议类似：该系统应该将思想分解成各个组成部分，从而使它们能够通过一小组基本符号的组合来表达。

As I mentioned in chapter 1, a number of seventeenth-century thinkers, including John Pell and Robert Hooke, envisioned a system that could express anything whatsoever with the clarity of algebraic symbols. Another proposal along these lines (although referring to a different type of symbol) came from René Descartes. In a 1629 letter to his friend Marin Mersenne, Descartes observes that the Hindi–Arabic numerals can express infinitely many numbers using only ten symbols, compounded together according to simple rules. “In a single day,” he writes, “one can learn to name every one of the infinite series of numbers, and thus to write infinitely many words in an unknown language. The same could be done for all other words necessary to express all the other things which fall within the purview of the human mind.”14 Leibniz’s universal characteristic was, at least in some versions, similar to this proposal: the system was supposed to analyze ideas into components and so enable them to be expressed through combinations of a small set of basic symbols.

尽管笛卡尔的提议具体提到了数字，但这些想法符合一种语言学而非数学的传统——创造一种有时被称为“哲学语言”的东西，一种比人们现在使用的语言更精确的人工交流形式。15这种想法源于奥卡姆的威廉等中世纪思想家，但在 17 世纪经历了一次重大复兴。16扬·阿莫斯·夸美纽斯、塞浦路斯·金纳和弗朗西斯·洛德威克在 17 世纪 40 年代尝试建立这样的系统。17 世纪60年代初，阿塔纳修斯·基歇尔和约翰·约阿希姆·贝歇尔进行了进一步的尝试，莱布尼茨称他们受到了启发。17大约在同一时间，艾萨克·牛顿也进行了尝试。然而，最完善的方案应归功于约翰·威尔金斯，他是一位科学博学者，最终成为英国圣公会主教，参与了伦敦皇家学会的创立。学者们普遍忽视了威尔金斯对莱布尼茨的重要性，事实上，他们的目标不同：威尔金斯主要试图促进交流，而莱布尼茨则希望创造一种能够产生新知识的发现方法。莱布尼茨直到构思出普适性之后才接触到威尔金斯的体系，因此它不能被视为至关重要的影响。但威尔金斯的著作是一个有用的比较点，因为作为该流派中一个异常详尽的范例，它揭示了莱布尼茨所参与的思想运动中一些被误解的方面。虽然莱布尼茨的计划通常被比作形式逻辑的后期发展，但它存在于17世纪的环境中，与我们如今所理解的形式主义相去甚远。对于莱布尼茨的同时代人来说，批判术语不是正式的；它是人为的。

Although Descartes’s proposal had specific reference to numerals, such ideas fit into a tradition that was more linguistic than mathematical—the idea of creating what is sometimes called a “philosophical language,” an artificial form of communication more precise than the languages people presently speak.15 This idea has roots in medieval thinkers such as William of Ockham, but it underwent a major resurgence in the seventeenth century.16 Jan Amos Comenius, Cyprian Kinner, and Francis Lodwick tried to build such systems in the 1640s. In the early 1660s, Athanasius Kircher and Johann Joachim Becher made further attempts, both of which Leibniz cites as inspirations.17 Around the same time, Isaac Newton tried his hand at it as well. The most full-fledged scheme, however, was attributable to John Wilkins, a scientific polymath and (eventually) Anglican bishop who was involved in the founding of the Royal Society of London. Scholars have generally dismissed the importance of Wilkins for Leibniz, and it is true that their aims were different: Wilkins was mainly trying to facilitate communication, whereas Leibniz wanted to create a method of discovery that could produce new knowledge. Leibniz did not encounter Wilkins’s system until after he conceived the universal characteristic, so it cannot be considered a crucial influence. But Wilkins’s work is a useful point of comparison because, as an uncommonly detailed example of the genre, it illustrates some misunderstood aspects of the intellectual movement in which Leibniz was participating. While Leibniz’s project is habitually likened to later developments in formal logic, it existed within a seventeenth-century milieu that was worlds away from formalism as we now understand it. For Leibniz’s contemporaries, the critical term was not formal; it was artificial.

威尔金斯在其1641年出版的著作《水星，或秘密而迅捷的信使》中首次探讨了“普遍字符”的概念。在解释了弗朗西斯·培根关于真实字符的概念之后，威尔金斯列举了四种能够被多种语言使用者理解的符号：数字、占星符号、化学符号和音符。然后，他以此方式探讨了“一种为表达其他一切事物而发明的通用书写形式”的可能性。18不同语言使用者对这些符号的发音会有所不同，就像占星符号一样在意大利语（或者也可以说，西班牙语）中发音为toro ，在英语中发音为bull；“但对每个人来说，意思都是一样的”。19虽然学者们经常将这一提议描述为“通用语言”，但威尔金斯明确表示，他将其视为一种新的书写形式，而不是一种语言。他写道：“完善这样一种发明，是统一第一次语言混乱的七十二种语言的唯一途径。” 20其目的不在于取代人们已经使用的语言；而在于通过一种不受巴别塔倒塌后强加于人类的语言混乱影响的方式在纸上记录思想，从而修复这些语言之间的裂痕。

Wilkins first discussed the idea of a universal character in his 1641 book Mercury, or the Secret and Swift Messenger. After explaining Francis Bacon’s idea of a real character, Wilkins lists four examples of symbols that could be understood by speakers of multiple languages: numerals, astrological symbols, chemical symbols, and musical notes. He then considers the possibility of “a generall kinde of writing invented for the expression of every thing else” in this manner.18 The symbols would be pronounced differently by speakers of different languages, as the astrological symbol is pronounced toro in Italian (or, one might add, Spanish) and bull in English; “yet the sense would be still the same” for everyone.19 While scholars have often characterized this proposal as a “universal language,” Wilkins makes it clear that he views it as a new form of writing, not a language. “The perfecting of such an invention,” he writes, “were the only way to unite the seventy two Languages of the first confusion.”20 The point is not to replace the languages people already speak; it is to repair the rifts between those languages by recording ideas on paper in a way that is unaffected by the Confusion of Tongues imposed on humankind after the fall of the Tower of Babel.

威尔金斯与他的朋友塞思·沃德详细讨论了这个想法，沃德在其 1654 年出版的《学院辩护》一书中发表了自己对这个问题的看法。对沃德来说，体现普遍性的最佳模型是符号代数，它能完整地呈现复杂的思想，同时避免“词语幻想造成的混乱或干扰”。21沃德驳斥了对手关于语法学家应该研究代数符号的建议，并坚决否认符号构成了语言：似是而非的算术“完全是针对语言及其仆从语法而设计的，其目的甚至在于反对使用数字本身，而学者们则将数字称为Lingua Mathematicorum（数学语言）。” 22也就是说，代数学家用a s 和x s 代替数字，以消除印地语-阿拉伯数字中残留的语言痕迹。虽然这句话有时被引用作为这一时期数学与语言对立的证据，但文章的其余部分却表明并非如此。23沃德继续说道：“符号的使用不仅限于数学，还应用于事物的本质”；他列举了毕达哥拉斯学派、卡巴拉学派和乔瓦尼·皮科·德拉·米兰多拉的组合学派作为例子。24沃德以“符号作家”的名义汇集的传统包括许多很快启发莱布尼茨的人物。25他赞美的既不是数学，也不是语言；而是符号，沃德小心地将符号与这两者区分开来。

Wilkins discussed this idea in detail with his friend Seth Ward, who published his own thoughts on the matter in his 1654 book Vindiciae academiarum (Vindications of Academies). For Ward, the best model for a universal character was symbolic algebra, which presented ideas in their full complexity while avoiding the “confusion or perterbati[on] of the fancy made by words.”21 Rejecting an opponent’s suggestion that grammarians should study algebraic symbols, Ward stridently denies that the symbols constitute a language: specious arithmetic “was a designe perfectly intended against Language and its servant Grammar, and that carried on so farre, as to oppose the use of numbers themselves, which by the Learned, are stiled Lingua Mathematicorum [the language of mathematics].”22 Algebraists, that is, replaced numbers with as and xs as a way to expunge the linguistic traces that persisted in Hindi–Arabic numerals. While this statement is sometimes cited as evidence of an opposition between mathematics and language in the period, the rest of the passage suggests otherwise.23 “The use of Symbols,” Ward continues, “is not confined to the Mathematicks only, but hath been applied to the nature of things”; he lists the Pythagoreans, cabbalists, and the ars combinatoria of Giovanni Pico della Mirandola as examples.24 The tradition Ward assembles under the name “Symbolicall writers” includes many of the figures who would soon inspire Leibniz.25 What he is celebrating in them is neither mathematics nor language; it is symbols, which Ward is careful to distinguish from both.

沃德接着提议将这种象征风格扩展为一个雄心勃勃的项目，类似于威尔金斯此前描述过的项目。在首次了解了“由韦达发明、哈里奥特推进、奥特雷德先生和笛卡尔完善的象征方法”后，沃德写道：“我产生了一个热切的愿望，希望同样的方法也能应用于其他事物。” 26这样做将提供一个系统，在这个系统中，“可以以论证的方式进行精确的论述，而不需要像在似是而非的分析操作中那样费力。 ” 27尽管沃德几乎肯定不知道笛卡尔写给梅森的信，但他所描述的内容却非常相似：将思想分解成简单的元素，这些元素可以用一小组易于学习的符号来表示。沃德认为，符号的优势在于它们使人们“能够自由地进行讨论……而不受文字的困扰，而当人类的思维被设计来表达时，它既不能如此清晰地看到结果，也不能像摆脱这些障碍时那样迅速地进行比较。” 28将文字描述为“障碍”是当时科学界态度的典型特征：语言似乎是一种烦恼，是阻碍人类知识进步的枷锁。

Ward goes on to propose extending this symbolical style into an ambitious project somewhat like the one Wilkins had previously described. After first learning about “the Symbolicall way, invented by Vieta, advanced by Harriot, perfected by Mr Oughtred, and Des Cartes,” Ward writes, “I was put upon an e[ar]nest desire, that the same course might be taken in other things.”26 Doing so would provide a system in which “exact discources may be made demonstratively without any other paines then [sic] is used in the operations of specious Analyties.”27 Although Ward was almost certainly unaware of Descartes’s letter to Mersenne, what he describes is quite similar: breaking ideas down into simple elements that could be represented by a small set of easily learned symbols. The advantage of symbols, according to Ward, is that they enable people “to discourse . . . freely without the trouble of words, upon which while the mind of man is intended, it neither sees the consequence so cleerely, nor can so swiftly make comparison as when it is acquitted of those obstacles.”28 The description of words as “obstacles” is characteristic of the attitudes of the scientific circles of the time: language appears as an annoyance, a shackle on humanity that holds back the advancement of knowledge.

沃德并没有认真尝试创造一种通用字符，但威尔金斯最终与乔治·达尔加诺和约翰·雷等人合作完成了这项工作。29 1668年，他出版了《关于一种真正的字符和一种哲学语言的文章》，该书长达 450 多页，是有史以来最详细的通用字符方案之一。威尔金斯的项目包含两部分：通过书面符号表达思想的“真正的字符”以及旨在取代英语等口头语言的“哲学语言”。30真正的字符由根据规则组装的特殊符号组成（图 2.1）。这些符号应该“任何国家都能用自己的语言辨认”，就像所有中国居民都使用相同的书写系统，每个人都“用自己的语言阅读” 。31另一部分，即哲学语言，旨在为现有语言提供一种替代品，它以基于拉丁字母和希腊字母松散的拼音字母印刷。这个系统与现有语言的不同之处在于，它是在某个创始时刻“根据艺术规则”形成的，而不是像英语和拉丁语那样经过漫长而偶然的过程。32因此，这些符号（根据威尔金斯的说法）将很容易被每个人理解，而且一旦确立使用，就能抵御文字容易产生的腐败。

Ward did not seriously attempt to create a universal character, but Wilkins, working together with collaborators such as George Dalgarno and John Ray, eventually did.29 In 1668, he published An Essay towards a Real Character and a Philosophical Language, which, at over 450 folio pages, is one of the most detailed universal character schemes ever produced. Wilkins’s project involved two parts: a “real character” that expressed ideas through written symbols, along with a “philosophical language” that was supposed to replace spoken languages like English.30 The real character consists of special symbols assembled according to rules (figure 2.1). These symbols are supposed to be “legible by any Nation in their own Tongue” just as all of the inhabitants of China use the same writing system, each “reading it in his own Language.”31 The other part, the philosophical language, is meant to provide a substitute for existing languages and is printed in a phonetic alphabet loosely based on Latin and Greek characters. This system is supposed to differ from existing languages in that it was formed “according to the rules of Art” at a single founding moment rather than emerging through a long, haphazard process like the ones that produced English and Latin.32 The symbols would thus (according to Wilkins) readily become comprehensible to everyone and, once established in use, resist the corruption to which words are liable.

威尔金斯体系的核心是一个详细的、分层的分类方案（图 2.2）。威尔金斯写道，这种层次结构“对需要标记或名称的事物或概念进行了公正的列举和描述”，其排列方式受到亚里士多德范畴体系和彼得·拉姆斯图表法的启发。33每个符号都表示该层次结构中的一组特定坐标（图 2.3）。还可以在字符上添加环和钩，以指示词性或动词变位，并以各种方式修改含义。威尔金斯还为代词和语法词提供了较小的字符；字符之间的相对位置也在确定其含义方面发挥作用。哲学语言的词语表达的内容或多或少（尽管正如詹姆斯·道格尔·弗莱明所说指出，并不完全是）相同的信息，映射到音节而不是形状上。34正如威尔金斯所解释的那样，这个系统的目的部分是为了澄清科学交流，部分是为了通过以一种（据称）任何语言的使用者都能理解的形式呈现宗教教义来协助传教工作。

The heart of Wilkins’s system is a detailed, hierarchical classification scheme (figure 2.2). This hierarchy, Wilkins writes, provides “a just Enumeration and description of such things or notions as are to have Marks or Names assigned to them,” arranged in a fashion inspired by the Aristotelian system of categories and Peter Ramus’s diagramming method.33 The symbols each indicate a particular set of coordinates within this hierarchy (figure 2.3). Loops and hooks can also be added on to characters to indicate part of speech or conjugation and to modify the meaning in various ways. Wilkins also provides smaller characters for pronouns and grammatical words; the way the characters are positioned relative to each other also plays a role in determining their meanings. The words of the philosophical language express more or less (although, as James Dougal Fleming has pointed out, not exactly) the same information, mapped onto syllables rather than shapes.34 The purpose of this system, as Wilkins explains it, is in part to clear up scientific communication and in part to aid missionary work by presenting religious doctrines in a form that could (supposedly) be understood by speakers of any language.

从我们的角度来看，威尔金斯的体系似乎与莱布尼茨最终创建的类似代数的体系几乎没有共同之处。威尔金斯体系的整体结构基于语法，而语法（沃德肯定会提醒我们）是一门与代数完全不同的学科。此外，威尔金斯并没有像莱布尼茨所建议的那样，将思想分解成简单的组成部分，而是采用了自上而下的分类法。但威尔金斯的工作在使用真实字符（即（据称）可以用多种语言阅读的视觉非语音符号）方面，确实类似于代数符号。威尔金斯对这些符号的明显执着长期以来一直困扰着语言思想史学家。正如雅普·马特所观察到的，真正的字符似乎没有必要，因为有了语音字母，哲学语言就可以同时用于书面和口头交流。35汉斯·阿尔斯莱夫和迈克尔·伊瑟曼都曾解释威尔金斯体系的双重性质，他们认为威尔金斯相信符号拥有捕捉事物真实本质的“神秘”力量。36但这种解释与他对此事的明确陈述相冲突。在书的最后一部分，他声称这些符号是“按制度”而不是“自然”表示的——这清楚地表明他认为真正的字符是人们出于务实考虑而任意选择的符号。37那么，他为什么要花费如此多的精力来创建单独的视觉符号和语音符号呢？

From our perspective, Wilkins’s system may seem to have little in common with the algebra-like system that Leibniz would ultimately create. The overall structure of Wilkins’s system is based on grammar, which (as Ward would surely remind us) was an entirely separate discipline from algebra. Moreover, rather than analyzing ideas into their simple components as Leibniz proposed, Wilkins employed a top-down taxonomy. But Wilkins’s work does resemble algebraic notation in its use of real characters—visual, nonphonetic symbols that can (supposedly) be read in multiple languages. Wilkins’s evident fixation on these symbols has long perplexed historians of linguistic thought. As Jaap Maat has observed, the real character appears to be unnecessary, since, with its phonetic alphabet, the philosophical language can serve for both written and spoken communication.35 As a way of explaining the dual nature of Wilkins’s system, Hans Aarsleff and Michael Isermann each have argued that Wilkins believed symbols to have a “mystical” power to capture the true natures of things.36 But this interpretation conflicts with his explicit statements on the matter. In the last section of the book, he asserts that the symbols signify “by Institution” rather than “Naturally”—a clear indication that he saw the real characters as arbitrary signs chosen pragmatically by people.37 Why, then, did he go to so much effort to create separate visual and phonetic symbols?

如果我们认识到，威尔金斯所处的文化在很大程度上仍属于口头文化，那么这一选择就更有意义了。在解释语言的真正特征时，威尔金斯借鉴了一种长期存在的思想传统，这种传统认为语言主要就是讲话，而字母书写则是对讲话的一种编码。威尔金斯沿袭了亚里士多德在《论诠释》中的符号理论，指出“名称”是人们一致同意用来表示“他们的心理概念”的“任意声音或词语”；他继续说：“书面文字是那种声音的图形或图像。” 38在分类表中，他将“阅读”归类为“特殊类型的说话”之一；他将其定义为指代“我们眼前所见的词语”的各种清晰的声音。39阅读通常需要说话的能力，这种观点在当时很普遍，从人们讨论耳聋后果的方式中就可以推断出来：人们普遍认为，如果不先学习口语，就无法学会阅读，尽管约翰·沃利斯不同意这种观点。40真正的字符有望通过使书写完全独立于言语来改变这种情况，就像数字可以向任何语言的使用者传达相同的含义一样。

This choice makes more sense if we recognize that he was working within what was still, to a large extent, an oral culture. In explaining the real character, Wilkins drew on a long-standing intellectual tradition that thought of language primarily as speech and viewed alphabetical writing as an encoding thereof. Following Aristotle’s theory of signs in On Interpretation, Wilkins states that “Names” are “arbitrary sounds or words” that people have agreed to use as signs of “their Mental notions”; “The Written word,” he continues, “is the figure or picture of that Sound.”38 In the taxonomic tables, he classifies “READING” as one of the “particular kinds of speaking”; he defines it as a variety of articulate sound that refers to “such words as we see before us.”39 The idea that reading ordinarily involved the ability to speak was common at the time, as one can infer from the way people discussed the consequences of deafness: it was generally presumed that people could not learn to read without first learning a spoken language, although John Wallis disagreed.40 Real characters promised to alter this situation by making writing fully independent of speech, much as numerals can convey the same meanings to speakers of any language.

这种解释仍然留下了一些关于真实字符应该如何运作的谜题。当然，尽管威尔金斯声称如此，但他的符号并不像数字那么容易学习。对于莱布尼茨来说，重要的不是威尔金斯计划的细节，而是使普遍字符的想法变得合理的一般思维方式。正如弗莱明所说，威尔金斯的项目以早期现代信仰为前提，即人类思维是世界的一面镜子，被赋予了与事物真实本质相符的想法。41 威尔金斯在《论文》的第一部分明确阐述了这一原则：“正如人们普遍同意相同的理性原则一样，他们也同样同意相同的内在概念或对事物的理解。” 42也就是说，诸如马和异教徒之类的概念已经存在于每个人的头脑中，等待着被发现。类似的假设也可以在威廉·奥特雷德关于通过代数方法将数学命题从“词语的隐蔽处”中“解放出来”的陈述中看出。符号。如果人们相信自己想要表达的想法已经在读者的脑海中等待着，那么，要想获得完美的清晰度，只需砍掉遮蔽理性之光的语言丛林即可。

This interpretation still leaves some puzzles regarding how the real character was supposed to work. Certainly, Wilkins’s symbols are not, in spite of his claims, nearly as easy to learn as numerals. With regard to Leibniz, what is important is less the specifics of Wilkins’s plan than the general way of thinking that made the idea of a universal character plausible. As Fleming has argued, Wilkins’s project is premised on the early modern belief that the human mind is a mirror of the world, divinely outfitted with ideas that suit the true natures of things.41 Wilkins states this principle explicitly in the first part of the Essay: “As men do generally agree in the same Principle of Reason, so do they likewise agree in the same Internal Notion or Apprehension of things.”42 That is, ideas such as horse and heretic are already there in every human mind, waiting to be uncovered. A similar assumption may be discerned in William Oughtred’s statement about “uncasing” mathematical propositions from the “covert of words” by means of algebraic symbols. If one believes that the ideas one wishes to express are already waiting in the reader’s mind, then all it takes to create perfect clarity is to cut away the verbal thickets that obscure the light of reason.

早在 1668 年威尔金斯的《论文》发表时，反对这种思维方式的声音就已经高涨。威尔金斯以前的合作伙伴乔治·达尔加诺提出了尖锐的批评。达尔加诺曾在该项目的早期阶段与威尔金斯合作过，但由于对系统设计的看法不同，两人最终分道扬镳。与沃德一样，达尔加诺认为将概念分解成各个部分比将它们置于一个层次结构中更好。他也对真实字符和语言之间的区别持怀疑态度。在他 1661 年出版的《符号的艺术》一书中，达尔加诺详细论证了“哑字符”与文字并不像他同时代人所认为的那样不同；“字符和声音的艺术，”他坚持认为，“是一回事”。43正如马特所说，达尔加诺的作品标志着人们朝着将语言视为抽象系统而非一套具体实践的理念迈出了重要一步，从而淡化了书写和口语之间的差异。44这种概念转变的含义是，书面符号并非天生就不受语言混乱的影响；由于它们依赖于任意意义，因此它们像英语和拉丁语一样，也受沟通偶然性的影响。

Opposition to this way of thinking was already mounting by the time Wilkins’s Essay was published in 1668. A pointed critique came from Wilkins’s erstwhile collaborator George Dalgarno. Dalgarno had worked together with Wilkins in the early phase of the project, but the two fell out as a result of a difference of opinion about how the system should be designed. Like Ward, Dalgarno thought it would be better to analyze concepts into parts rather than place them in a hierarchy. He also became skeptical of the distinction between real characters and languages. In his 1661 book The Art of Signs, Dalgarno argues at length that “mute characters” are not as different from words as his contemporaries tended to assume; “the art of characters and sounds,” he maintains, “is one and the same.”43 Dalgarno’s work, as Maat has argued, marks an important step toward the idea of language as an abstract system rather than a set of concrete practices, and thus to a de-emphasis of the difference between writing and speech.44 An implication of this conceptual shift was that written symbols were not inherently immune to the Confusion of Tongues; since they depended on arbitrary significations, they were subject to the contingencies of communication just as much as English and Latin were.

达尔加诺对语言和文字之间界限的抨击并不是真实文字观在莱布尼茨时代面临的唯一威胁。一种更具认识论色彩的批评思路源自笛卡尔传统。尽管笛卡尔在给梅森的信中表达了对普遍符号体系的想法，但他怀疑这一梦想能否实现。笛卡尔的哲学强调他所谓的清晰、独特的感知的重要性，即心灵能够感知并与其他感知截然不同的感知。45因此，笛卡尔在给梅森的信中得出结论，没有“真正的哲学”，即关于世界上一切事物的完整而正确的理论，就不可能创造出普遍的符号体系。46这一论点一旦被接受，便会抑制将数学符号的清晰度扩展到其他领域的想法。从笛卡尔的角度来看，一门科学只有建立在像算术一样确定的原理之上时，才能用数字的清晰度来表达；在此之前，普遍性还必须等待。

Dalgarno’s attack on the division between languages and characters was not the only threat that the real-character view faced in Leibniz’s time. A more epistemological line of criticism emerged from the Cartesian tradition. Although Descartes entertained the idea of a universal symbolism in his letter to Mersenne, he was skeptical that the dream could be realized. Descartes’s philosophy emphasized the importance of what he called clear and distinct perceptions, meaning perceptions that are accessible to the mind and sharply distinguished from other perceptions.45 As a result, Descartes concludes in his letter to Mersenne that a universal symbolism could not be created without “la vraie philosophie” (the true philosophy)—that is, a complete and correct theory of everything in the world.46 This argument, when accepted, put a damper on the idea of extending the clarity of mathematical symbols to other areas. From a Cartesian perspective, a science can only be expressed with the clarity of numerals when it rests on principles as certain as those of arithmetic; until then, the universal character would have to wait.

我们将会看到，笛卡尔的“真正哲学”（vraie philosophie）论证在符号方法的讨论中反复出现。那些试图将这些方法的威力扩展到新领域的人一次又一次地面临批评，认为他们的想法不够清晰。然而，莱布尼茨至少在他职业生涯的早期，莱布尼茨认为笛卡尔的反对意见是可以克服的。诚然，莱布尼茨对语言与字符的区别的关注程度不如威尔金斯；他有时将普遍特征描述为一种“语言”，尽管这种语言的运作方式应该与文字不同。但他赞同威尔金斯的信念，即心灵与现实之间存在着神圣的对应关系。与他的英国前辈一样，他将语言问题主要定位于能指的排列，并假设一个设计合理的符号系统对每个人都是透明的。莱布尼茨相信他可以两者兼得，既接受我们看来的形式主义，又坚持符号具有概念意义，这正是他的方法与我们所知的算法思维的区别所在——简而言之，这也是他不现代的原因。

As we will see, Descartes’s “vraie philosophie” argument would recur frequently in discussions of symbolic methods. Again and again, people who attempted to extend the power of these methods to new domains would face the criticism that their ideas were insufficiently clear. Yet Leibniz, at least early in his career, thought that the Cartesian objection could be overcome. To be sure, Leibniz made less of the language–character distinction than Wilkins did; he at times described the universal characteristic as a type of “language,” albeit one that was supposed to work differently from words. But he shared Wilkins’s belief in a divinely instituted correspondence between mind and reality. Like his English predecessor, he located the problems of language primarily in the arrangement of signifiers, presuming that a properly designed system of symbols would readily become transparent to everyone. Leibniz’s confidence that he could have it both ways, embracing what looks to us like formalism while still maintaining that the symbols had conceptual meanings, is what differentiates his approach from algorithmic thinking as we know it—in short, what makes him not modern.

深入事物内部

Into the Interior of Things

莱布尼茨最早尝试建立哲学语言的时间早于威尔金斯的《论文》。他在青少年时期撰写的《组合艺术论文》（1666 年）探讨的是现在所说的组合学：即确定简单元素可以有多少种组合方式的问题，例如ABC （如果我们忽略顺序）可以以三种方式组合成对：AB、AC或BC。47他借用梅森的符号，将这些组合称为“com2nations”；当三个元素组合在一起时，它们就构成了“ con3nations”，依此类推。48他对这种方法的期望很广泛。莱布尼茨希望将组合学变成一门通用的“艺术”，可以应用于从神学到法学等广泛的领域。这项工作标志着他毕生项目的开始，并最终促成了他关于通用计算方法的著名论述。

Leibniz’s earliest attempts at a philosophical language predate the publication of Wilkins’s Essay. The Dissertation on the Art of Combinations (1666), which he wrote while still a teenager, deals with what is now called combinatorics: the problem of determining how many ways simple elements can be compounded together, as ABC can (if we disregard order) be combined into pairs in three ways: AB, AC, or BC.47 Borrowing a notation from Mersenne, he called these assemblages “com2nations”; when three elements are combined, they make up “con3nations,” and so forth.48 His ambitions for this method are broad. Leibniz wanted to turn combinatorics into a general “art” that could be applied to a wide range of fields, from theology to jurisprudence. This work marks the beginning of the lifelong project that would eventually lead to his famous statements about a universal method of calculation.

“组合艺术”的概念通常可以追溯到中世纪思想家拉蒙·卢尔。49卢尔于1232年左右出生于马略卡岛，30 岁时获得了一种精神幻象，这促使他抛弃家庭，将余生奉献给福音传道。为此，他发展了所谓的“大艺术”（ Ars Magna），这是一种普遍的推理方法，据称涵盖逻辑和形而上学。这种方法的一些版本涉及一种名为转盘的可旋转装置，它可以生成特定符号集的所有组合；其他版本则涉及复杂的表格和图表。使用卢尔的艺术，人们可以根据某些固定术语迭代所有可能的问题和陈述，例如“永恒的巨大差异是什么？”和“什么是伟大而永恒的一致性？” 50例如卢尔的这一努力源于宗教：他希望利用自己的艺术创作论据，使人们信服基督教的真理。然而，卢尔后来的崇拜者却将他的思想用于各种目的。例如，约翰·海因里希·阿尔斯泰德在编纂其1630年的百科全书时就运用了卢尔的思想，而莱布尼茨后来也以此为起点，开始了自己百科全书的创作。

The idea of an “art of combinations” is often traced back to the medieval thinker Ramon Llull.49 Born around 1232 in Majorca, Llull had a spiritual vision at the age of thirty that led him to abandon his family and devote the rest of his life to evangelism. To this end, he developed what he called the Ars Magna (Great Art), a general method of reasoning that is supposed to encompass both logic and metaphysics. Some versions of this method involved a rotatable device called a volvelle that could generate every combination of a certain set of symbols; others involved intricate tables and diagrams. Using Llull’s art, one can iterate through all possible questions and statements based on certain fixed terms, such as, “What is great difference of eternity?” and “What is great and eternal concordance?”50 For Llull, the point of this endeavor was religious: he intended to use his art to produce arguments that convinced people of the truth of Christianity. Llull’s later admirers, however, turned his ideas to a variety of ends. Johann Heinrich Alsted, for instance, employed Llullian ideas in the organization of his 1630 encyclopedia, which Leibniz later used as a starting point for his own encyclopedic endeavors.

在《博士论文》中，莱布尼茨试图将鲁尔的艺术扩展为一种普遍的“发明逻辑”——一种遵循弗朗西斯·培根《学术的进展》精神发现新知识的方法。51该方法始于识别未定义的原始术语，并用数字表示。他将这种数值表示法与霍布斯“推理即计算”的论述联系起来。52他还引入了一种简化方法，将复杂的概念表示为伪分数：是组合序列中的第三个，按特定顺序枚举。53

In the Dissertation, Leibniz attempts to extend the Llullian art into a general “Logic of invention”—a method for discovering new knowledge in the spirit of Francis Bacon’s Advancement of Learning.51 The method begins with the identification of primitive terms that are to be left undefined, which he represents by numbers. He links this numerical representation to Hobbes’s statement that reasoning is computation.52 He also introduces a shorthand by which complex ideas may be expressed as pseudofractions: is the third in the sequence of com2nations, as enumerated in some particular order.53

莱布尼茨尝试使用这种符号来开发一种列举所有可能逻辑推理的方法。作为这个系统如何运作的例子，他列出了 27 个原始术语，然后从中推导出一些定义。例如，他这样定义无限。54相关的原始术语如下： 9表示部分，11 表示相同，14 表示数字，15 表示复数，17 表示可能，18 表示全部，19 表示给定。要从这里到达无限，我们必须遵循一系列复杂度不断增加的组合。首先，他定义：“数量是 14 个 9（15）。”接下来是 ：“相等，A为 11，。”（也就是说，数量相同。）然后是：“当A中的 9大于B时，它就更大。” （请注意，这意味着相等，而不是三分之二！）最后，我们可以理解的定义：“无限，比18、19、17大。”虽然这些定义包括一些将事物联系在一起的拉丁词，但莱布尼茨认为，通过将语法类别简化为简单的组成部分，就可以完全通过数字所代表的原始术语来定义新的想法。

Using this notation, Leibniz attempts to develop a method for enumerating all possible logical inferences. As an example of how this system would work, he lists twenty-seven primitive terms and then derives some definitions from them. For example, here is how he defines infinite.54 The relevant primitive terms are as follows: 9 is part, 11 is same, 14 is number, 15 is plurality, 17 is possible, 18 is all, and 19 is given. To get from here to infinite, we must follow a series of combinations of increasing complexity. First, he defines : “Quantity is 14 of 9s (15).” Next, : “Equal, A of 11, .” (That is to say, something of same quantity.) Then, : “A is Greater when it has a 9, more than B.” (Note that means equal, not two-thirds!) Finally, we can understand the definition of : “Infinite, than 18, 19, 17.” While these definitions include some Latin words linking things together, Leibniz suggests that, by reducing grammatical categories to their simple components, it would be possible to define new ideas entirely through primitive terms represented by numbers.

虽然他最初将该系统作为生成三段论的一种手段，但他随后提出了另一个从该系统“像推论一样流淌出来”的想法：开发“一种通用符号 [ Scriptura Universalis ]，即无论读者使用何种语言，都能理解的符号”的可能性。55他将这种符号描述为一种可与埃及象形文字和汉字相媲美的“通用多字符号”。56莱布尼茨的多字符号应该由表示原始术语的图形符号组成，例如表示数字的点组和表示关系的线；这些符号将提供“一种字母表”，可以从中组装出更复杂的术语。57结果将是“一种通用文字，它将像“它具有通用性，无需借助字典即可阅读，并能让人立即掌握所有事物的基础知识。” 58用简单组件构建符号的想法类似于笛卡尔、沃德和达尔加诺的提议，他们都更喜欢这种分析方法，而不是威尔金斯的分类方法。其结果就是他后来所说的“普遍特征”，即一套其构造反映了事物真实本质的符号系统。

Although he initially presents this system as a means of generating syllogisms, he then proceeds to another idea that “flows like a corrolary” from it: the possibility of developing “a Universal Notation [Scriptura Universalis], that is, intelligible to whatever reader, no matter what his language.”55 He describes this notation as a “universal polygraphy” comparable to the Egyptian hieroglyphs and the Chinese characters.56 Leibniz’s polygraphy is supposed to consist of pictorial symbols for primitive terms, such as groups of points for numbers and lines for relations; these symbols would provide “a kind of alphabet” out of which more complex terms could be assembled.57 The result would be “a universal script that will be as easy as it is general, readable without the aid of a dictionary, and allowing fundamental knowledge of everything to be absorbed at once.”58 The idea of constructing symbols from simple components resembles the proposals of Descartes, Ward, and Dalgarno, who all preferred this analytical approach to Wilkins’s taxonomic one. The result would be what, later in his life, he would come to call a “universal characteristic,” a system of symbols whose construction reflected the true natures of things.

莱布尼茨后来将这篇论文视为一部少年作品，但他在整个职业生涯中仍在不断反思普遍写作的理念。在1677年的论文《迈向普遍特征》中，他回顾了当时他对这一项目的思考发展历程。他写道，早期他希望创造一种“人类思想的字母表”，以便将组合艺术应用于其中：“通过字母的连接和对由字母组成的单词的分析，其他一切都可以被发现和判断。” 59后来，他放弃了字母模型，转而采用用“特征数”来表示思想的想法，这些特征数将基于类似于百科全书的综合知识体系来确定。60这些数字将使形而上学和道德成为可能，使其服从于“一种绝对可靠的计算方法”。61他用另一句经常被引用的与计算相关的陈述来总结其后果：“一旦为大多数概念建立了特征数，人类将拥有一种新的工具，它将比光学仪器增强眼睛的程度更大地增强思维能力，并将取代显微镜和望远镜，就像理性优于视觉一样。” 62正如他在这里所描述的，这个项目比威尔金斯的项目雄心勃勃得多。莱布尼茨并不想创造一种用于国际交流的固定语言；相反，他想创造一种可以用来推进科学知识的发现方法。

Leibniz later came to view the Dissertation as a piece of juvenilia, but he continued to revisit the idea of a universal writing throughout his career. In the 1677 essay “Towards a Universal Characteristic,” he recounts the development of his thinking on the project up to that point. Early on, he writes, his hope was to create an “alphabet of human thoughts” to which the combinatorial art could be applied: “through the connection of its letters and the analysis of words which are composed out of them, everything else can be discovered and judged.”59 More recently, he had moved away from the alphabetical model toward the idea of representing ideas by “characteristic numbers,” which would be determined based on a comprehensive system of knowledge akin to an encyclopedia.60 These numbers would make it possible to subject metaphysics and morals to “an infallible method of calculation.”61 He sums up the consequences in another statement that is often quoted in relation to computation: “Once the characteristic numbers are established for most concepts, mankind will then possess a new instrument which will enhance the capabilities of the mind to a far greater extent than optical instruments strengthen the eyes, and will supersede the microscope and telescope to the same extent that reason is superior to eyesight.”62 This project, as he describes it here, is far more ambitious than that of Wilkins. Leibniz did not want to produce a fixed language for use in international communication; instead, he wanted to create a method of discovery that could be used to advance scientific knowledge.

莱布尼茨在1677年至1678年间的努力展现出他对特征数的痴迷，近乎命理学。1677年，他开始撰写一篇关于威尔金斯真实性格的详细评论，这或许可以被视为他试图构建实现该特征所必需的百科全书式知识。63评论的大部分内容是威尔金斯层级结构中概念的拉丁定义；莱布尼茨也曾短暂地尝试将一个简单的定义转化为数字形式，批评了威尔金斯的一些假设，并构建了他自己层级结构的框架版本。1678年，他两次尝试构建一个特征数，但都以失败告终，一次名为“通用语言”（Lingua Generalis），另一次名为“通用语言”（Lingua Universalis）。可能受到达尔加诺先前工作的启发，他构建了一套在文字和数字之间转换的规则：

Leibniz’s efforts from 1677 and 1678 exhibit an obsession with characteristic numbers that verges on numerology. In 1677, he began writing a detailed commentary on Wilkins’s real character that might be seen as an attempt to construct the sort of encyclopedic knowledge that would be necessary to realize the characteristic.63 Much of the commentary consists of Latin definitions for concepts in Wilkins’s hierarchy; Leibniz also briefly tried to convert a simple definition into a numerical form, criticized some of Wilkins’s assumptions, and constructed his own skeletal version of the hierarchy. In 1678, he made two abortive attempts at a characteristic, one titled “Lingua Generalis” and the other “Lingua Universalis.” Probably inspired by the previous work of Dalgarno, he constructs a set of rules for converting between words and numbers:

这个词由辅音-元音对构成，辅音表示数字，元音表示数字的位置，顺序任意；因此，用他的例子来说，“ bodifalemu ”这个词的数字是 81374。其思想是通过乘法来表示思想的复合；数学语句“ 48 est 6 ^{narius ”——也就是说，48 可以被}6整除——因此可以翻译成逻辑语句“人类是理性的”。65或者，按照莱布尼茨的规则，我们可以说fema is ha。

The word is constructed from consonant–vowel pairs, with the consonant indicating the digit, the vowel indicating its place, and the order being arbitrary; thus, to use his example, the word “bodifalemu” has the number 81374. The idea is to represent the compounding of ideas through multiplication; the mathematical statement “48 est 6^narius”—that is, 48 is divisible by 6—could thus, for instance, translate into the logical statement a human is rational.65 Or, by Leibniz’s rules, one could say that fema is ha.

莱布尼茨在这两篇文本中的工作展现了他对语言音乐和声的密切关注。正如他所解释的那样，他的体系不赋予音节顺序任何意义，因此人们可以随意选择书写bodifalemu或mubodilefa。他写道，这种灵活性使“令人愉悦、恰当的语言能够回归音乐和诗歌，以及所有其他令人愉悦的话语之中。” 66莱布尼茨在第二个残篇中更进一步，声称一行诗句可以用这种语言创作，“就像通过一个确定的演示”，“一切都已确定”。67虽然路易·库图拉引用这段话来表明莱布尼茨甚至在美学领域也借鉴了数学方法，但应该注意到，对严谨性的追求与 17 世纪的诗学并不陌生。68人们尤其会想到德国巴洛克诗人，例如马丁·奥皮茨，他于 1624 年出版的《德国诗歌集》确立了一种强调押韵格式、严格的诗体和语言纯粹性的诗歌风格；其要点是在严格的限制条件下，展现出对词汇组合的精湛技艺。69莱布尼茨的《博士论文》广泛讨论了具有组合方面的诗歌形式，包括尤利乌斯·凯撒·斯卡利杰所说的多变诗体——其各部分可以按多种顺序排列的诗体——以及回文结构。70例如，莱布尼茨计算出斯卡利杰的下面这行诗可以重新排列成 64 种方式：

Leibniz’s work in these two texts exhibits a close attention to the musical harmonies of language. As he explains, his system gives no significance to the order of syllables, so that one can indifferently choose to write either bodifalemu or mubodilefa. This flexibility, he writes, enables a “pleasing, apt language to be restored to Music and poetry, and to all other delights of discourse.”66 Leibniz goes further in the second fragment, claiming that a line of verse could be composed in this language “as if by a sure demonstration” with “everything determined.”67 While Louis Couturat takes this passage to show that Leibniz drew on mathematical methods even in the domain of aesthetics, it should be noted that the desire for rigor is not alien to seventeenth-century poetics.68 One thinks, in particular, of German baroque poets such as Martin Opitz, whose 1624 Book of German Poetry codifies a poetic style that emphasizes rhyme schemes, rigid verse forms, and the purity of language; the point is to undertake a virtuosic display of assembling words under heavy constraints.69 Leibniz’s Dissertation contains extensive discussions of poetic forms that have a combinatoric aspect, including what Julius Caesar Scaliger had called protean verses—verses whose parts can be arranged in multiple orders—and palindromes.70 For instance, Leibniz calculates that the following line by Scaliger can be rearranged in sixty-four ways:

这类做法在当时确实很普遍，反映了当时人们对“艺术”的普遍评价，即技艺。莱布尼茨所处的文化比后来的文化更重视遵守规则，而且在更广泛的领域也是如此；至少在莱布尼茨的一些他的工作中某些方面在我们看来像是数学化的，在当时是理所当然的事情。

Such practices were, indeed, widespread in the period, reflecting the general valuing of “art” in the sense of artifice. Leibniz was working within a culture that valued the following of rules to a greater extent and in a wider range of fields than it would be valued in later periods; at least some of the aspects of his work that look to us like mathematization would have been, at the time, matter of course.

在评价莱布尼茨在计算史上的意义时，牢记这一文化背景至关重要。普遍性通常被视为类似于符号逻辑或编程语言；在这两种解释中，莱布尼茨都认为符号的形式关系优先于其含义。但他的目标比现代逻辑或计算机科学的目标更为广泛。其目的不仅在于提供处理符号的规则，还在于建立一个通用的符号系统，为包括法律、音乐、形而上学和神学在内的广泛领域提供论证确定性。在1678年写给沃尔特·冯·奇恩豪斯的一封信中，莱布尼茨将这一特征描述为卡尔达诺和韦达代数方法的推广，用于处理“与量级毫无共同之处的其他公式”。72他向持怀疑态度的朋友保证，这种方法不会失去其现实基础：“没有人应该担心对字符的思考会让我们远离事物本身；相反，它会引导我们进入事物的内部。” 73这个充满信心的声明提出了一个问题：特征数字应该如何与符号应该表达的思想和事物相关联。

This cultural context is important to keep in mind while evaluating Leibniz’s significance in the history of computation. The universal characteristic is often viewed as akin to either symbolic logic or a programming language; in both interpretations, Leibniz is taken to privilege formal relations of symbols over their meanings. But his goals are wider than those of either modern logic or computer science. The point was not just to provide rules for what to do with symbols but also to establish a universal sign system that would grant demonstrative certainty to a wide range of domains, including law, music, metaphysics, and theology. In a 1678 letter to Walter Von Tschirnhaus, Leibniz describes the characteristic as a generalization of the algebraic methods of Cardano and Viète to work with “other formulas which have nothing in common with magnitude.”72 He assures his skeptical friend that this method will not lose its grounding in reality: “No one should fear that the contemplation of characters will lead us away from the things themselves; on the contrary, it leads us into the interior of things.”73 This confident statement raises the question of how the characteristic numbers are supposed to relate to the ideas and things that the symbols were supposed to express.

在这方面，与威尔金斯的比较颇具启发性。尽管莱布尼茨显然遵循着与英国主教不同的方案，但两人在他们明显不关心的事情上却有相似之处——即词语在确立符号意义中的作用。在《随笔》的开篇，威尔金斯声称，先前提出真实字符方案的作者“在最初的奠基过程中普遍犯了错误”，因为他们试图“根据某种特定的语言，从一本词典中构建出这样一个字符，而没有参考事物的本质”。74威尔金斯的字符不应该编码英语；它应该编码在任何语言出现之前就存在于头脑中的纯粹的观念。然而，分类表仍然是由英语词汇构建的，没有英语词汇，分类表就毫无意义。威尔金斯似乎完全不关心词汇在系统构建中的使用，这正是前洛克时代对语言态度的典型特征，正是这种态度使得真实字符的理念成为可能。对于威尔金斯而言，概念的清晰度并非源于如何解释符号，而是源于整个系统按照“艺术规则”排列，从而反映了对世界的科学理解。

In this regard, the comparison with Wilkins is illuminating. Although Leibniz is clearly following a different plan from the English bishop, the two are alike in what they are conspicuously not concerned about—namely, the role of words in establishing the meanings of symbols. At the beginning of the Essay, Wilkins claims that previous authors of real-character schemes “did generally mistake in their first foundations” because they attempted “the framing of such a Character from a Dictionary of Words, according to some particular Language, without reference to the nature of things.”74 Wilkins’s character is not supposed to encode the English language; it is supposed to encode bare ideas that exist in the mind prior to any language. But the classification tables are nonetheless built from English words, without which they would be meaningless. That the use of words in the construction of the system did not seem to bother Wilkins at all is characteristic of the pre-Lockean attitudes toward language that made the real-character idea possible. As far as Wilkins is concerned, conceptual clarity stems not from how the symbols are explained but from the fact that the system as a whole is arranged according to “the rules of art” and thus reflective of a scientific understanding of the world.

就意义问题而言，莱布尼茨的立场更类似于威尔金斯，而非现代形式主义。他并没有将沟通意义的问题排除在他的符号体系之外，而是假设设计得当的能符会迅速向所有人揭示其含义。在给茨尔恩豪斯的信中，他提出了一个（回顾起来）值得怀疑的主张，类似于威尔金斯在《人类学论文》中提出的主张：该系统将像数字一样易于学习。基于普遍特征，莱布尼茨写道：“也可以发展一种口语和书面语……它可以在几天内学会，足以表达日常实践中发生的一切，并在批评和发现方面具有惊人的价值，就像数字字符一样。” 75他相信这种语言比英语等更容易学习，源于这样一种信念：语言混乱主要源于“我们使用的字符排列不当”。76有了设计得当的特征，我们“只需要看到字符，就能自由而毫不费力地将足够的概念带入我们的大脑。” 77 人们的假设是，学习一门通用语言并不需要学习一套新的观念，也不用学习一整套看待世界的方式。“概念”早已存在于头脑中，等待被感知；人们需要做的只是学习符号。

As far as the issue of meaning goes, Leibniz’s position is more akin to that of Wilkins than to modern formalism. Rather than exclude issues of communicational meaning from his symbolic system, he presumed that properly designed signifiers would disclose their meanings promptly to everyone. In the letter to Tschirnhaus, he makes a (retrospectively) questionable claim similar to one Wilkins makes in the Essay: that the system will be as easy to learn as numerals. On the basis of the universal characteristic, Leibniz writes, “a spoken and written language can also be developed . . . which can be learned in a few days and will be adequate to express everything that occurs in everyday practice, and of astonishing value in criticism and discovery, after the model of the numeral characters.”75 His confidence that this language would be so much easier to learn than, say, English stems from a belief that linguistic confusion results primarily from the fact that “the characters we use are badly arranged.”76 With a properly designed characteristic in place, we would “need merely to see the characters in order to have adequate notions brought to our mind freely and without effort.”77 The assumption is that learning a universal language does not involve learning a new set of ideas, a whole way of thinking about the world. The “notions” are already there in the mind, waiting to be perceived; all one need do is learn the symbols.

莱布尼茨从未放弃特征数的理念，但后来他的主张变得更加慎重。例如，他将语言学研究的精力转向发展简化版拉丁语；因此，他在逻辑演算方面的工作与他试图建立通用语言的努力背道而驰。78他对特征数所能实现的目标的期望也随之减弱。他关于这些问题最成熟、尽管常常被误解的论述之一出现在1685年的《发现的艺术》中。在这篇文章中，莱布尼茨探讨了一种艺术的可能性，这种艺术“在其他事物上能够完成类似于代数对数字所做的事情” 。79他对比了实现这一目标的两种途径。第一种是建立一个如此完美的符号系统，以至于它包含了所有可能知识的种子。“如果词语能够按照一种我认为可行的、但那些构建通用语言的人尚未发现的手段来构建，”他写道，“那么我们就能通过词语本身达到预期的结果，这将对人类生活产生不可思议的效用。” 80然而，需要强调的是，莱布尼茨在此描述这一理念，是为了承认笛卡尔对此的反对意见。莱布尼茨总结道，这个梦想在理论上是可能的，但如果没有一个完整而正确的万物理论——也就是说，没有笛卡尔的“真正的哲学”（vraie philosophie），它就无法实现。

Leibniz never gave up on the idea of characteristic numbers, but he later grew somewhat more measured in his claims. For instance, he switched his linguistic efforts toward developing a simplified version of Latin; his work on logical calculi thus diverged from his attempts at a universal language.78 His ambitions for what the characteristic could do also contracted. One of his most mature, albeit often misunderstood, statements on these issues appears in the 1685 “Art of Discovery.” In this essay, Leibniz discusses the possibility of an art that would “accomplish in other matters something similar to what Algebra does with numbers.”79 He contrasts two approaches to this goal. The first is a system of signs so perfect that it contains the seeds of all possible knowledge. “If words were constructed,” he writes, “according to a device that I see possible, but which those who have built universal languages have not discovered, we could arrive at the desired result by means of words themselves, a feat which would be of incredible utility for human life.”80 It is important to stress, however, that Leibniz is describing this idea here so as to concede to the Cartesian objection against it. The dream, Leibniz concludes, is theoretically possible, but it could not be achieved without a complete and correct theory of everything—that is, without Descartes’s “vraie philosophie.”

莱布尼茨为实现这一遥不可及的梦想而提出的替代方案，与其说是一台计算机，不如说是一种检查工作成果的方法。“与此同时，”他继续说道，“另一条不那么优雅的道路已经摆在我们面前，而另一条道路则必须完全重新构建。” 81另一种方法涉及“像数学家那样，利用合适的字符来固定我们的想法，并为其添加数值证明。” 82这句话可能被认为表明了类似符号逻辑的东西，尽管它没有（而且正如我们现在所知，不能）提供纯机械的程序来回答任何问题，但它确实允许以一种允许机械地检查其正确性的形式来表达证明。但莱布尼茨的解释表明了其他的东西。他在文章前面指出，数学的确定性源于这样一个事实：“我们可以不断地试验或检验不仅可以得出结论，而且可以在任何时候，通过将整体简化为数字，从前提中迈出的每一步。” 83他继续说道，正是由于缺乏这样的“测试”，其他领域才不如数学那么确定。正是在这种背景下，我们找到了那句著名的“让我们计算一下”的段落。这种背景表明，莱布尼茨所想的类似于程序员所说的“健全性检查”：为了测试 3 是否是y ³ = 8 y + 3 的根，人们将其代入公式，并验证 3 ³确实等于 8 · 3 + 3。

Leibniz’s alternative to this unattainable dream is less like a computing machine than a method of checking one’s work. “In the meantime,” he continues, “there is another less elegant road already open to us, whereas the other would have to be built completely new.”81 This other approach involves “making use as mathematicians do, of characters, which are appropriate to fix our ideas, and of adding to them a numerical proof.”82 This statement might be taken to indicate something like symbolic logic, which, although it does not (and, as we now know, cannot) provide a purely mechanical procedure for answering any question, does allow proofs to be expressed in a form that allows their correctness to be checked mechanically. But Leibniz’s explanation suggests something else. The certainty of mathematics, he states earlier in the essay, stems from the fact that “we can continually submit to trials or tests not only the conclusion but also, at any moment, each step made from the premises, by reducing the whole to numbers.”83 It is because of the lack of such “tests,” he continues, that other fields are not as certain as mathematics. It is in this context that we find the famous “let us calculate” passage. This context suggests that what Leibniz has in mind is something like what programmers call a “sanity check”: to test that 3 is a root of y³ = 8y + 3, one plugs it in and verifies that 3³ does, in fact, equal 8 · 3 + 3.

事实上，这种测试是当时计算实践的一个常见元素。正如我在第一章中提到的，早期的算法文本通常包含特殊的程序，有时称为“证明”或“证明”，旨在检查计算结果。例如，加法的一个“证明”就是所谓的“去九”过程。将加数的各个数字相加，当总数超过九时减去九，对和也这样做。如果这些数字不相等，那么和一定是错的。莱布尼茨在他1677年版的“让我们计算”语句中明确提到了这个程序：

This sort of test was, indeed, a common element of the computational practices of the period. As I mentioned in chapter 1, early algorism texts often included special procedures, sometimes called “proofs” or “proves,” designed to check the results of calculations. One “proof” of addition, for instance, was the procedure called casting out nines. One adds up the individual digits of the addends, subtracting nine whenever the total exceeds nine, and does the same for the sum. If these numbers are unequal, then the sum must be wrong. Leibniz had referenced this procedure explicitly in the 1677 version of his “Let us calculate” statement:

此外，我们应该能够让世人信服我们本应发现或得出的结论，因为验证计算结果很容易，只需重新计算，或者尝试类似算术中去掉九的测试即可。如果有人怀疑我的结果，我会对他说：“先生，我们来算一下吧。”这样，我们就能很快拿起笔墨解决问题。84

Moreover, we should be able to convince the world what we should have found or concluded, since it would be easy to verify the calculation either by doing it over or by trying tests similar to that of casting out nines in arithmetic. And if someone would doubt my results, I should say to him: “Let us calculate, Sir,” and thus by taking to pen and ink, we should soon settle the question.84

如果这就是他所说的“数值证明”，那么他所提出的与其说是一种现代意义上的逻辑演算（结论的确定性源于正式的推理规则），不如说是一种旨在揭示错误的实用方法。莱布尼茨无疑知道，“去九”程序并不总是有效；当结果偏差为九的倍数时，它就无法检测到错误。因此，其重点不在于用机械过程取代推理来确保正确性；而是在于“检验论证”，正如他在1677年的文章中所写，从而揭示其缺陷。85

If this is what he means by “numerical proof,” then what he is proposing is less a logical calculus in the modern sense, in which the certainty of conclusions stems from formal inference rules, than a pragmatic measure meant to reveal errors. As Leibniz undoubtedly knew, the casting out nines procedure does not always work; it fails to detect errors when the result is off by a multiple of nine. The point, then, is not to ensure correctness by replacing reasoning with a mechanical process; it is “to examine arguments,” as he writes in the 1677 essay, and thus reveal their flaws.85

我们可以从莱布尼茨的言论中看出类似的务实态度关于计算机器。在与《发现的艺术》同一年撰写的一份经常被引用的手稿中，他描述了自己制造这台机器的用途：“优秀的人不应该像奴隶一样浪费时间进行计算，如果使用机器，这些时间可以安全地交给其他人。” 86虽然人们常常认为他提到奴隶制是为了通过计算自动化将人们从工作中解放出来，但这是对这段文字的误读。这台机器的目的是确保计算能够“安全地”分配给下属，而下属大概不能被信任用纸笔正确地完成计算。与数值“测试”一样，其目的是弥补人类容易犯错的倾向；莱布尼茨在文中前面写道：“从那些尝试求圆的方位的失败中可以看出，算术是几何精确性的最可靠守护者。” 87通过使计算变得更容易、更可靠，机器将鼓励天文学家和其他“优秀人士”通过数值测试来检查他们的结果，从而提高知识的可靠性。

A similarly pragmatic attitude may be discerned in Leibniz’s remarks about calculating machines. In an oft-quoted manuscript written the same year as “The Art of Discovery,” he describes the purpose of the machine he built: “it is unworthy of excellent men to lose hours like slaves in the labor of calculation, which could be safely relegated to anyone else if the machine were used.”86 While this reference to slavery has often been taken to indicate that he means to liberate people from work by automating calculation, this is a misreading of the passage. The point of the machine is to ensure that calculations can be “safely” assigned to a subordinate who, presumably, cannot be trusted to do them correctly with a pen and paper. As in the numerical “tests,” the aim is to compensate for the human tendency to err; “it is known,” Leibniz writes earlier in the text, “from the failures [of those] who attempted the quadrature of the circle that arithmetic is the surest custodian of geometrical exactness.”87 By making calculation easier and more reliable, the machine would encourage astronomers and other “excellent men” to check their results through numerical tests and thus improve the reliability of knowledge.

如果莱布尼茨在 17 世纪 80 年代放弃了他对这一特征的一些更极端的说法，那么他仍然相信符号的力量。在撰写《发现的艺术》之后的几年里，他对他曾在《博士论文》中承诺的理性演算进行了一些最复杂的尝试。其中最著名的尝试出现在两份手稿中，这些手稿有时是关于被称为加减演算的。88这个想法是将算术加法推广为“实加法”，正如他在第二份手稿中所说的那样，它处理的是实体的集合而不是抽象的数量。89正如他所解释的那样，在常见的算术中 2 + 2 = 4，但这只有在假设被组合的两对数字不同时才成立；如果一个人有两枚硬币，并将同样的两枚硬币相加，那么他仍然只有两枚硬币。90实加法应该表示加法如何与可能像这样重叠的集合一起工作。尽管他之前的努力是通过将字母放在一起（ra表示理性动物）或通过素数来实现的，但该系统采用了类似于代数 + 和 − 的运算符，从而在逻辑和代数之间建立了明显的关系。

If Leibniz was moving away from some of his more extreme claims about the characteristic in the 1680s, he retained his faith in the power of symbols. In the years following the writing of “The Art of Discovery,” he developed some of his most sophisticated attempts at the rational calculus he had promised in the Dissertation. The best known of these appear in a pair of manuscripts about what is sometimes called the plus–minus calculus.88 The idea is to generalize arithmetical addition into a “real addition,” as he called it in the second manuscript, that deals with collections of entities rather than abstract quantities.89 As he explains, 2 + 2 = 4 in common arithmetic, but this only holds if the two pairs being combined are assumed to be different; if one has two coins and adds the same two coins, one still has only two coins.90 Real addition is supposed to represent how addition works with collections that may overlap like this. Whereas his previous efforts had worked either through placing letters together (ra means rational animal) or through prime numbers, this system employed operators similar to the algebraic + and −, creating an apparent relation between logic and algebra.

在该系统的第一个版本中，莱布尼茨使用A + B = L来表示A包含在L中，B是余数。他还定义了一个互补关系A – B ，表示从A中减去B后剩下的部分。莱布尼茨的系统有点像数值代数，尽管并不完全相同：如果A和B包含某些共同点，则A + B – B ≠ A ，因为减法会“消除” A中包含的某些东西。91在第二份手稿中，他省略了减法，专注于实数加法，现在他将其与通过在加号周围画一个圆圈来完成数值加法。他对这个运算提出了两种不同的解释。在大多数情况下，他认为它表达了属性的组合，因此R ⊕ A可能表示既有理性又有动物性的东西。他顺便指出，这个运算也可以表达多个物种组成一个属的关系，因此（举个例子）F ⊕ B 可能表示一个同时包含鱼和鸟的类。92他根据一组公理推导出这个系统的结论，其中包括A ⊕ A = A，这后来成为布尔逻辑的核心定律。

In the first version of the system, Leibniz uses A + B = L to indicate that A is contained in L, with B being the remainder. He also defines a complementary relation A – B, which expresses whatever is left over when B is taken away from A. Leibniz’s system works somewhat like numerical algebra, although it is not exactly the same: if A and B contain something in common, then A + B – B ≠ A, since the subtraction “destroy[s]” something that is contained in A.91 In the second manuscript, he leaves out subtraction and focuses on real addition, which he now distinguishes from numerical addition by drawing a circle around the plus sign. He suggests two distinct interpretations of this operation. In most cases, he takes it to express the combination of attributes, so that R ⊕ A might signify what is both rational and animal. The operation can also, he notes in passing, express the relation by which multiple species compose a genus, so that (to invent an example) F ⊕ B might signify a class containing both fish and birds.92 He works out the consequences of this system based on a set of axioms, including A ⊕ A = A, which would later become a central law of Boolean logic.

莱布尼茨逻辑演算的遗产早已经过了晚年一系列的重读和修订。虽然一些评论家暗示乔治·布尔的体系是在莱布尼茨奠定的基础上建立的，但这种直接影响的说法并没有历史证据支持；布尔在发展他的逻辑体系时可能并不知道莱布尼茨的相关著作。93 他后来回忆说，当布尔终于在19世纪 50 年代读到莱布尼茨的逻辑著作时，他得出结论，这位莱比锡哲学家表达他的观点的“语言对于那些接受了后来对逻辑功能更为公正的看法的人来说一定显得夸张”。94 1900年左右，伯特兰·罗素试图将莱布尼茨打造成一个逻辑形式主义者，他认为莱布尼茨的普遍特征“显然类似于布尔实践的现代符号逻辑科学”；路易·库图拉 (Louis Couturat)、C.I. 刘易斯 (C.I. Lewis) 和恩斯特·卡西尔 (Ernst Cassirer) 扩展了这一观点。95但这些解读必须围绕莱布尼茨思想中违反现代逻辑学科界限的方面进行。

The legacy of Leibniz’s logical calculi has long been filtered through a series of rereadings and revisions that took place in much later periods. While some commentators have intimated that George Boole built his system on foundations laid by Leibniz, the claim of a direct line of influence is not supported by historical evidence; Boole was probably unaware of Leibniz’s relevant writings when he developed his logical system.93 When Boole finally did read Leibniz’s logical work in the 1850s, he later recalled, he concluded that the Leipzig philosopher had stated his views “in language which to those imbued with later and juster views of the functions of Logic must appear extravagent.”94 Around 1900, Bertrand Russell attempted to make a logical formalist of Leibniz, arguing that the universal characteristic “was evidently akin to the modern science of Symbolic Logic” as practiced by Boole; Louis Couturat, C. I. Lewis, and Ernst Cassirer expanded on this view.95 But these readings had to work around aspects of Leibniz’s thought that violate the disciplinary bounds of modern logic.

一个特别的难点在于莱布尼茨如何处理那个反对意见（经常针对经典演绎逻辑提出），即结果仅仅是定义的重述，因此最终是任意的。在一份早期手稿中，莱布尼茨回应道：“即使某些命题被任意假设，如术语的定义，但从中产生的真理远非任意的。” 96作为例子，他给出了数字系统，它由任意定义的符号组成，但仍然得出绝对真理的结论。这个论点在关注符号关系方面预见了形式主义；它类似于编程语言如何将名称视为可互换意义上的任意名称。然而，形式主义在其现代形式中并不允许建立莱布尼茨想要在这些符号和物质世界之间建立的那种联系。莱布尼茨提出的加减运算并不只是一种推导出公理结果的方法：这些符号应该表达关于事物本质的真理，使人们能够仅通过计算就得出结论，从而为其他学科所做的就如同印度计算技术为数量所做的一样。

A particular difficulty regards how Leibniz addresses the objection (often raised against classical deductive logic) that the results were mere restatements of definitions and were thus ultimately arbitrary. In an earlier manuscript, Leibniz responds that, “even if certain propositions are arbitrarily assumed, as the definitions of terms, yet there arises from these a truth which is far from arbitrary.”96 As an example, he gives the numeral system, which consists of arbitrarily defined symbols but which nonetheless produces conclusions of absolute truth. This argument anticipates formalism in its focus on the relations of symbols; it resembles how programming languages treat names as arbitrary in the sense of interchangeable. Yet formalism does not, in its modern form, license the sort of connection Leibniz wanted to make between such symbols and the material world. Leibniz did not present the plus–minus calculus as a mere way of working out the consequences of axioms: the symbols were supposed to express truths about the natures of things that would enable one to draw conclusions about them through calculation alone, thus doing for other subjects what the Indian calculating techniques did for quantities.

莱布尼茨坚信这一点是可能的，只有当我们考虑到他思想诞生的十七世纪的严峻考验时，才说得通。包括威尔金斯和莱布尼茨在内的普遍性特征的倡导者都假定，在人们学习任何语言之前，头脑中就存在一套普遍的观念。莱布尼茨在1686年的《形而上学论述》中明确阐述了这一观点：“我们不可能学到任何我们头脑中没有的事物，观念就像形成思想的物质一样。” 97人类头脑中已经具备观念的假设意味着，一旦为特定的知识领域开发出适当的符号系统，任何足够聪明的人都可以轻松地理解这些符号。这种相信符号透明性的理由与任何严肃的文化或语言相对主义都不相容。正如威尔金斯的真实性格和奥特雷德版本的似是而非的算术一样，莱布尼茨的特点是使用符号作为任意的能指，但如果人们承认思想也可以是任意的，以取决于人们成长过程中偶然情况的方式组合在一起，那么梦想就会立即从人们的掌握中退出。

Leibniz’s confidence that this was possible makes sense only if we consider the seventeenth-century crucible from which his thinking flowed. Advocates of universal characters, including Wilkins and Leibniz alike, presumed the existence of a universal set of ideas that exist in the mind before one learns any language. Leibniz states this position explicitly in the 1686 Discourse on Metaphysics: “there is nothing we could ever learn of which we do not already have in our mind the idea, which is like the matter out of which the thought is formed.”97 The assumption that the human mind comes already outfitted with ideas entailed that, once an adequate symbolism was developed for a given area of knowledge, the symbols would readily become comprehensible to anyone intelligent enough. This rationale for trusting in the transparency of symbols is incompatible with any serious sort of cultural or linguistic relativism. As with Wilkins’s real character and Oughtred’s version of specious arithmetic, Leibniz’s characteristic employs symbols as arbitrary signifiers, but if one admits that ideas, too, can be arbitrary, put together in ways that depend on the contingent circumstances of one’s upbringing, the dream promptly withdraws from grasp.

在莱布尼茨的早期追随者中，莱布尼茨在特征研究方面最卓有成效的部分是对逻辑演算的尝试。莱布尼茨最杰出的追随者之一（继孜孜不倦的克里斯蒂安·沃尔夫之后）是瑞士逻辑学家约翰·海因里希·兰伯特。他从18世纪50年代开始尝试发现“隐藏在莱布尼茨特征和组合学中的秘密”。98兰伯特开发了一个基于类似维恩图的表示法的逻辑系统，正如他所提出的，该系统旨在对质进行代数符号对量的处理。他的目标是创建一个科学的符号系统，该系统能够如此准确地表征世界，以至于“事物理论和符号理论变得可以互换”。99兰伯特在1767 年出版的《论题》一书中介绍了他对这种科学符号的尝试。100然而，他所发展的东西并没有达到他所创造的期望；《论题》的早期读者失望地发现，他的逻辑演算仅仅使用字母来表示性质，而不是像他所承诺的那样，使用与事物本质自然对应的符号。101尽管兰伯特付出了努力，但一个可以仅通过符号就产生有关各种主题的真实知识的系统的梦想仍然没有实现。

Among Leibniz’s early followers, the most fruitful parts of his work toward the characteristic were the attempts at a logical calculus. One of Leibniz’s most prominent acolytes (after the tireless Christian Wolff) was the Swiss logician Johann Heinrich Lambert, who attempted, starting in the 1750s, to discover what “was concealed in the Leibnizian characteristic and in the ars combinatoria.”98 Lambert developed a logic system based on a representation somewhat like Venn diagrams, which was meant, as he presented it, to do for qualities what the algebraic notation had done for quantities. The goal was to produce a scientific sign system that represents the world with such veracity that, he writes, “the theory of things and the theory of signs become interchangeable.”99 Lambert presents his attempt at such a scientific symbolism in his book Disquisitio, published in 1767.100 What he developed, however, did not live up to the expectations he had created; early readers of the Disquisitio were disappointed to find that his logical calculus merely used letters to represent qualities rather than, as he had promised, symbols that naturally corresponded to the essences of things.101 In spite of Lambert’s efforts, the dream of a system that can produce true knowledge about all manner of subjects through symbols alone remained unrealized.

如果说莱布尼茨的这一特征只是空谈，那么莱布尼茨在另一个领域——数学——取得了巨大的成功。他版本的无穷小微积分很大程度上得益于他精心设计的符号，例如dx和∫ dx；虽然莱布尼茨的符号体系饱受诟病，但它几乎普遍被认为优于牛顿。正如下一节所示，莱布尼茨在微积分方面的工作促成了17世纪末期围绕符号方法的论述的重新构建。他的符号创新证明了，在某种程度上，实数符号的倡导者是正确的：符号确实可以做到文字无法做到的事情。但莱布尼茨的工作也提出了新的概念问题。解释例如“+”符号的含义很容易，但解释dx的含义却不那么容易。因此，批评者质疑莱布尼茨的方法是否仅仅是工具性程序，达不到欧几里得和阿基米德所设定的论证知识的标准。如果说，莱布尼茨在其关于普适特征的研究中坚信符号会牢牢地依附于普适的“概念”，那么他在微积分方面的工作则动摇了这种信念，提出了符号即使毫无意义也能产生真实结果的可能性。

If the characteristic turned out to be vaporware, Leibniz was a massive success in another area: mathematics. His version of the infinitesimal calculus draws much of its power from his carefully designed symbols, such as dx and ∫dx; while the Leibnizian notation has its detractors, it is almost universally viewed as superior to that of Newton. As the next section shows, Leibniz’s work on calculus contributed to a reframing of the discourse surrounding symbolic methods around the end of the seventeenth century. His notational innovations offered proof that, in a way, advocates of real characters were right: symbols really could do things that words could not. But Leibniz’s work also raised new conceptual problems. It was easy enough to explain what, say, the + sign meant, but explaining the meaning of dx was not so easy. Critics thus questioned whether Leibniz’s methods were merely instrumental procedures that fell short of the standards of demonstrative knowledge set by Euclid and Archimedes. If, in his work on the universal characteristic, Leibniz maintained confidence that symbols would remain firmly affixed to universal “notions,” his work on calculus troubled that faith, raising the possibility that symbols could produce true results even though they meant nothing at all.

可以说是一种算法

An Algorithm, So to Speak

弗洛里安·卡乔里 (Florian Cajori) 在其全面的数学符号史中指出，只有两位数学家能够创造出两个以上流行的符号：莱布尼茨和莱昂哈德·欧拉。102尽管莱布尼茨没有像几十年前的皮埃尔·埃里戈内 (Pierre Hérigone) 那样将符号主义发挥到极致——他没有试图完全取代文字——但他确实在他的数学实践中赋予了符号突出的地位。他的符号观的核心是盲思观念。在《论文》中，莱布尼茨指出，在一次智力活动中可以同时思考许多事情，“例如，通过阅读纸上的数字，我们在一种盲目的思考中掌握了一个非常大的数字，玛土撒拉所有的年龄都不足以明确地数出来。” 103关键在于，人们不需要对一百万个事物有一个清晰的心理图像，就能正确地推理出这个数量——人们只需要理解符号 1000000 是如何构造的，以便用莱布尼茨喜欢说的话来“表达”这个数字。104

In his comprehensive history of mathematical notations, Florian Cajori states that only two mathematicians ever managed to create more than two symbols that caught on: Leibniz and Leonhard Euler.102 Although Leibniz did not take symbolism to the extreme that Pierre Hérigone had a few decades before—he did not attempt to replace words altogether—he did give symbols a prominent role in his mathematical practice. Central to his view of notation is the idea of blind thought. In the Dissertation, Leibniz states that it is possible to think of many things at once in a single intellectual act, “as, for example, when by reading numerals on a piece of paper we grasp in a kind of blind thought some very large number that all the years of Methuselah would not suffice to count up explicitly.”103 The point is that one need not have a clear mental image of a million things in order to reason correctly about that quantity—one need only understand how the symbol 1000000 is constructed so as to “express,” as Leibniz liked to put it, the number.104

尽管莱布尼茨将其盲目思考的理念应用于我们如今视为算法的一些实践，但将其与现代形式的算法思维等同起来是错误的。盲目思考并不意味着盲目遵循规则；它是一种沉思，而不是一种缺乏智能的机械过程。105莱布尼茨不仅使用符号来计算结果，还将其用于以表格形式可视化数学数据，从而使模式可见。106在一篇导致人们对莱布尼茨在计算机发展中所扮演的角色大加赞赏的文本中，我们可以找到这种方法的清晰例子。众所周知，莱布尼茨对二进制表示法很感兴趣。1703 年，他在《皇家科学院论文集》上发表的一篇文章中介绍了二进制数字系统，并展示了如何仅使用 1 和 0 进行基本的算术运算。他认为二进制算术解释了《易经》的含义，并认为它可能在汉字的形成过程中也发挥了一定作用，他推测这些汉字可能揭示了“一些关于数字和思想的相当重要的东西”。107他在文章结尾处作了一条富有启发性的评论，将这些思想与普遍特性联系起来，即“人们从概念中得出的任何推理都可以通过某种计算方式从其字符中得出” 。108

Although Leibniz applied his idea of blind thought to some practices that we would now see as algorithmic, it would be a mistake to identify it with algorithmic thinking in its modern form. Blind thought did not mean blind rule following; it was a form of contemplation, not an unintelligent, mechanical process.105 Leibniz used symbols not just for computing results but also for visualizing mathematical data in a tabular form so as to make patterns visible.106 One may find a clear example of this approach in one of the texts that has led to outsize claims about Leibniz’s role in the development of computers. Leibniz was, as is well known, interested in binary notation. In an article published in the journal Memoires de l’Académie Royale des Sciences in 1703, he introduces the binary numeral system and shows how the basic operations of arithmetic can be performed with only 1 and 0. He argues that binary arithmetic explains the meaning of the I Ching and suggests that it may have played some role in the formation of the Chinese characters as well, speculating that these characters might reveal “something considerable about numbers and ideas.”107 He ends the article with a suggestive comment linking these ideas to the universal characteristic, in which “every reasoning that one can draw from notions could be drawn from their characters by a manner of calculation.”108

莱布尼茨在本文中讨论的二进制系统实际上就是构成现代计算基础的二进制系统。但莱布尼茨对这个系统感兴趣的原因并不是它对计算机科学如此重要的原因。二进制算术之所以成为现代计算的核心，主要是因为它为电子计算机器奠定了便捷的基础。对于使用转鼓的机械计算器来说，进制越高越好，这就是莱布尼茨在他的计算机中使用十进制而不是二进制的原因。莱布尼茨对二进制的兴趣并非源于它在机械化中的实用性，而是源于它的视觉特性：它使数字中的模式更容易被看到，从而使得人们能够轻松感知十进制数中不易辨别的数学定律。例如，在二进制表中，人们“一眼就能看出双重几何级数的一个著名性质的原因”——即任何数字都可以由二的幂之和组成。109重点不在于开发一种新的计算方法，正如他在文章中指出的那样，最好还是坚持十进制系统。重点在于思考不同类型的符号能告诉我们关于数学实体的什么信息。

The binary system that Leibniz discusses in this article is indeed the same one that forms the basis of modern computation. But the reasons this system interested Leibniz are not the reasons it is important to computer science. Binary arithmetic became central to modern computation primarily because it forms a convenient basis for electrical computing machines. For mechanical calculators using rotating drums, higher bases were better, which is why Leibniz used base ten rather than binary for his calculating machine. Leibniz’s interest in binary stemmed not from its utility in mechanization, but from its visual qualities: making patterns in numbers easier to see enables the easy perception of mathematical laws that cannot readily be discerned in decimal numbers. In a table of binary numbers, for instance, one “sees in a single glance the reason for a celebrated property of the double geometric progression”—namely, the fact that any number can be composed by summing powers of two.109 The point is not to develop a new method of computation, for the purpose of which, as he notes in the article, it would be best to stick with the base ten system. The point, rather, is to think about what different types of symbol can tell us about mathematical entities.

莱布尼茨在微积分研究中运用符号方法取得了最大的成功。研究过该领域的读者都知道，微积分关注的是两种运算，我们现在称之为积分和微分。积分的基本问题（在17世纪通常称为求积）是确定具有弯曲边界的形状（例如圆）的面积。在欧几里得几何中，没有人能够“化圆为方”，即构造一个与给定圆面积相同的正方形。数学家们很早就认识到，可以通过计算接近填满该面积的多边形的面积来近似该面积；多边形的边越多，近似值就越精确（图2.4）。波斯数学家贾姆希德·卡什（Jamshīd al-Kāshī）在15世纪初就曾使用这种方法来近似圆周率的值。110

Leibniz had the greatest success with symbolic methods in his work on calculus. As readers who have studied the field will know, calculus focuses on two operations that we now call integration and differentiation. The basic problem of integration (which was usually called quadrature in the seventeenth century) is to determine the area of a shape with a curved boundary, such as a circle. In Euclidean geometry, no one had ever been able to “square the circle,” meaning to construct a square with the same area as a given circle. Mathematicians had long recognized that it was possible to approximate this area by computing the areas of polygons that come close to filling it; the more sides the polygon has, the more accurate the approximation can be (figure 2.4). The Persian mathematician Jamshīd al-Kāshī had used such techniques to approximate the value of pi in the early 1400s.110

利用微积分方法，人们可以计算出（至少在很多情况下）当多边形的复杂度趋向无穷大时，这种近似值所趋近的值，从而精确地确定一个形状的面积。在牛顿和莱布尼茨的时代，这种方法通常用无穷小量来解释——这一说法引发了一些哲学问题，困扰了整个十八世纪的数学家们。微分是积分的逆运算，其目的是求函数在给定点的变化率（或者，等价地，曲线的斜率）。微分可以用来寻找曲线的最小点和最大点，这也是莱布尼茨特别感兴趣的问题。

With the methods of calculus, one can compute, at least in many cases, the value that this approximation approaches as the complexity of the polygon grows toward infinity, thus exactly determining the area of a shape. In the time of Newton and Leibniz, this method was often explained in terms of infinitely small magnitudes—a claim that raised philosophical problems that would occupy mathematicians through the eighteenth century. Differentiation, which is the inverse of integration, is about finding the rate of change of a function (or, equivalently, the slope of a curve) at a given point. Differentiation can be used to find the minimum and maximum points of curves, which was a problem that particularly interested Leibniz.

尽管牛顿曾指责莱布尼茨剽窃，但共识现在的问题是，两人独立发展了各自的微积分版本。牛顿的版本被他称为“流数法”，涉及流动数字的直觉；这是一个出了名的难以理解的理论。莱布尼茨的版本更接近今天的微积分教学方式，特别是在符号方面。他在 1684 年的文章中首次发表了这项工作，该文章专门关注微分。111微分一个方程就是把它转换成另一个方程，表示给定点的变化率。虽然存在一些重要的区别，但莱布尼茨的微分规则与现代微积分中使用的规则明显相似：例如x ^a变成ax ^{a −1}。112两年后，莱布尼茨在另一篇文章中发表了他的积分方法（他更喜欢称之为求和）；正是在这里，他引入了积分的符号∫ dx。积分符号应该类似于sum的s；该符号最初是印刷的，使用当时常见的s的长形式“ſdx” 。113

Although Newton famously accused Leibniz of plagiarism, the consensus now is that the two developed their versions of calculus independently. Newton’s version, which he called the “method of fluxions,” involved the intuition of flowing numbers; it is a notoriously difficult theory to understand. Leibniz’s version is closer to how calculus is taught today, especially in terms of its notation. He first published this work in the 1684 article, which focuses specifically on differentiation.111 To differentiate an equation is to transform it into another equation that indicates the rate of change at a given point. Although there are some important differences, Leibniz’s rules for differentiation are recognizably similar to the ones used in modern calculus: x^a, for instance, becomes ax^a−1.112 Leibniz published his method of integration (which he preferred to call summation) in another article two years later; it is here that he introduced the notation ∫dx for integrals. The integral sign is supposed to resemble an s for sum; the notation was originally printed, using the long form of s that was common at the time, as “ſdx.”113

虽然莱布尼茨和牛顿都使用符号记号，但他们对符号体系的理论意义持有不同看法。牛顿通常反对过度依赖符号的方法，而倾向于几何论证；在1715年发表于《哲学汇刊》的一篇关于他与莱布尼茨争论的文章中，牛顿以第三人称宣称：“牛顿先生并没有将他的方法置于符号形式中，也没有将自己局限于任何特定类型的流和流数符号。” 114另一方面，莱布尼茨则肆意发展新的符号记号，并在1684年的文章中提出了一些抽象的操作规则，他称之为“算法”，但并未提供任何关于该程序有效性的详细证明。现代学者对莱布尼茨是否成功为其微积分发展出严谨的理论基础持有不同看法；当然，他确实花费了大量时间思考理论问题。115但即使他确实有一套严谨的理论来支撑该方法，他发表的著作中并没有对此做出清晰的阐述，他与其他数学家的交流中也没有。他留给后人去苦苦思索这种新“算法”的概念基础。

While they both employed symbolic notations, Leibniz and Newton had different views on the theoretical significance of symbolism. Newton was generally opposed to methods that depended heavily on symbols, preferring geometric demonstration; in a 1715 Philosophical Transactions article about his dispute with Leibniz, Newton declared in the third person that “Mr. Newton doth not place his Method in Forms of Symbols, nor confine himself to any particular Sort of Symbols for Fluents and Fluxions.”114 Leibniz, on the other hand, developed new symbolic notations with abandon, and his 1684 article presented abstract rules of operation, which he called an “algorithm,” shorn of any detailed proof of the procedure’s validity. Modern scholars have taken varying views on whether Leibniz managed to develop a rigorous rationale for his calculus; certainly, he did spend a significant amount of time thinking about theoretical matters.115 But even if he did have a rigorous theory to back up the method, his published work did not contain a clear statement of it, nor did his correspondence with other mathematicians. He left his followers, instead, to puzzle over the conceptual foundations of the new “algorithm.”

莱布尼茨在其已发表的著作中倾向于回避基础性问题，这导致一些学者将他视为工具主义者。116但即使莱布尼茨的追随者们乐于向前推进，而不必为理论操心，当他们被迫证明其工作的合理性时，纲领性工具主义并非可行的选择。仅仅指出“算法”有效很难令人满意，因为只要没有可接受的有效性证明，它就仍然有可能在尚未经过检验的特定情况下失效。（正如我们将在第五章中看到的，（几个世纪后，这类问题成为计算机科学中算法理论发展的主要动力。）莱布尼茨的一些早期批评者，如米歇尔·罗尔，就是从这个角度攻击他的。117此外，无穷小的概念在莱布尼茨微积分教科书中（如纪尧姆·德洛必达和克里斯蒂安·沃尔夫的教科书中）发挥了重要的解释作用，但似乎包含着逻辑上的不可能性。118 莱布尼茨未能（至少在公开场合）清楚地说明他使用无穷小的理由，这导致批评者质疑他的方法是否仅仅是一些技巧，在严肃的科学中不应该存在。

Leibniz’s tendency to avoid foundational questions in his published work has led some scholars to view him as an instrumentalist.116 But even if Leibniz’s followers were content to plow ahead without fretting over theory, programmatic instrumentalism was not an available option when they were pressed to justify their work. Simply pointing out that the “algorithm” worked was hardly satisfactory since, as long as there was no acceptable proof of its validity, it remained possible that it would fail in particular cases that had as yet gone untested. (As we will see in chapter 5, such problems would, centuries later, become a major motivation for the development of a theory of algorithms in computer science.) Some of Leibniz’s early critics, such as Michel Rolle, attacked him from this angle.117 Further, the idea of the infinitesimal, which played an important explanatory role in Leibnizian calculus textbooks such as those of Guillaume de l’Hôpital and Christian Wolff, seemed to contain a logical impossibility.118 Leibniz’s failure to state (at least publicly) a clear rationale for his use of infinitesimals led critics to question whether his methods were mere tricks that had no place in serious science.

这一方面最著名的抨击来自爱尔兰哲学家乔治·贝克莱，他在其 1734 年的小册子《分析家：或，对一个异教徒数学家的演讲》中批评了牛顿和莱布尼茨的理论。贝克莱在文中提出了许多论点，其中包括指责微积分不够科学，因为它只提供解决问题的方法，而没有将问题置于坚实的概念基础上。贝克莱问道：“如果原理不科学，推理也不公正，那么是否就没有办法达到真理呢？这种方法应该被称为诀窍还是科学？” 119在贝克莱看来，微积分仅仅是一项“诀窍”，因为它只提供了解决问题的程序，而（在他看来）这些程序并没有建立在原理之上。这种批评不仅仅是试图表明莱布尼茨的“算法”可能会产生错误的结果。对于贝克莱来说，即使结果总是正确的，如果其概念基础存在缺陷，该程序仍然是不科学的；特别是，它可能包含两个相互抵消的错误。

The best-known attack on this front came from the Irish philosopher George Berkeley, who criticized the theories of both Newton and Leibniz in his 1734 pamphlet The Analyst: Or, a Discourse Addressed to an Infidel Mathematician. Berkeley makes a number of arguments in this text, among them the accusation that the calculus falls short of the status of a science because it merely provides methods for solving problems without placing them on solid conceptual ground. Berkeley asks, “Whether there be not a way of arriving at Truth, although the Principles are not scientific, nor the Reasoning just? And whether such a way ought to be called a Knack or a Science?”119 In Berkeley’s judgment, the calculus was a mere “knack” because it only provided procedures for solving problems without (in his view) founding those procedures on principles. This line of critique did more than try to show that Leibniz’s “algorithm” could produce wrong results. For Berkeley, even if the results were always right, the procedure could still be unscientific if its conceptual basis were flawed; in particular, it might contain two errors that canceled each other out.

许多学者指出，贝克莱反对微积分的论点与他早期关于算术和代数的著作之间存在明显的不一致，在这些著作中，他采取了毫不掩饰的工具主义立场。120在被称为《哲学评论》的早期笔记本中（可能写于 1707-8 年），贝克莱阐述了一种原始形式主义的算术和代数观点，其中“数字不过是名称，仅仅是词语”。121贝克莱对数字的解释至少在某些方面让人想起莱布尼茨的盲目思考观念。贝克莱认为，算术和代数是“纯粹语言的科学，除了在人类社会中实践外完全无用。它们没有推测性的知识，也没有思想的比较。” 122贝克莱认为，这些科学的价值恰恰在于这些名称，它们告诉我们一些无法在事物本身中立即感知到的东西：“告诉我有 10000 个人，比把他们都画出来给我看，能让我了解得更多，也知道得更多。” 123早些时候，他把这种言语科学的确定性与符号（不像贝克莱认为的观念）可以随意选择的事实联系起来：“我们之所以能如此出色地论证符号，是因为它们完全是任意的，在我们的掌控之中，可以随意创造。” 124

A number of scholars have pointed to an apparent inconsistency between Berkeley’s arguments against calculus and his earlier writings on arithmetic and algebra, in which he took an unapologetically instrumentalist stance.120 In the early notebooks known as the Philosophical Commentaries, probably written in 1707–8, Berkeley articulated a protoformalist view of arithmetic and algebra in which “Numbers are nothing but Names, meer Words.”121 Berkeley’s account of numbers is, at least in some aspects, reminiscent of Leibniz’s idea of blind thought. Arithmetic and algebra, Berkeley argues, are “sciences purely Verbal, & entirely useless but for Practice in Societys of Men. No speculative knowledge, no comparing of Ideas in them.”122 The value of these sciences, Berkeley suggests, consists precisely in these names, which tell us something that cannot immediately be perceived in things themselves: “I am better inform’d & shall know more by telling me there are 10000 men than by shewing me them all drawn up.”123 Earlier he links the certainty of such verbal sciences to the fact that symbols (unlike ideas in Berkeley’s view) may be chosen at will: “The reason why we can demonstrate So well about signs is that they are perfectly arbitrary & in our power, made at pleasure.”124

在某种程度上，这一论点与《分析家》的论战之间明显的分歧源于他攻击牛顿和莱布尼茨的政治动机。但是，贝克莱的工具主义代数理论转化为所有数学领域对符号操作的普遍接受，这一点在他的时代并不像在我们这个时代那么明显。牛顿和莱布尼茨都将微积分主要视为几何学的问题，而几何学作为示范科学的典范，在当时是神圣不可侵犯的，而代数则不然。正如道格拉斯·杰塞夫所强调的，贝克莱同样将微积分视为几何学的一部分，他对待几何学的方式与算术和代数截然不同。125牛顿本人在英文版的《通用算术》中指出：“乘法、除法以及诸如此类的计算是几何学新近引入的，这是不谨慎的，并且违背了这门科学的最初设计。” 126将符号方法应用于几何学违反了学科界限，并且没有被广泛接受的认识论框架来证明这种违反是合理的。

To an extent, the apparent disjunction between this argument and the polemics of The Analyst stemmed from his political motives in attacking Newton and Leibniz. But that Berkeley’s instrumentalist theory of algebra should translate into a general acceptance of symbol manipulation in all mathematical fields would not have been nearly as obvious in his time as it might be in ours. Newton and Leibniz both treated calculus as primarily a matter of geometry, which was, as the model demonstrative science, sacrosanct at the time in a way that algebra was not. As Douglas Jesseph has stressed, Berkeley likewise viewed calculus as a part of geometry, which he treated very differently from arithmetic and algebra.125 Newton himself stated, in the English edition of Universal Arithmetick, that “Multiplications, Divisions, and such sort of Computations, are newly received into Geometry, and that unwarily, and contrary to the first Design of this Science.”126 Applying symbolic methods to geometry was a violation of disciplinary boundaries, and there was no widely accepted epistemological framework in which this violation could be justified.

问题明显地出现在莱布尼茨决定将他的微分程序描述为一种“算法”时。这种用法出现在他 1684 年首次介绍该程序的文章等处：“根据这个规则，可以说它是这种微积分的算法，我称之为微分，所有其他微分方程都可以借助于一般微积分找到。” 127虽然莱布尼茨在这里确实将他的方法称为算法（Algorithmo），但他用“可以说”（ut ita dicam）来限定这种描述，表明这种用法并不十分标准。从他的其他著作中可以明显看出，他特指的算法是指与算术的类比。在 1692 年用法语撰写的一篇期刊文章中，莱布尼茨将微积分描述为“一种新的算法 [ algorithme ]，也就是说，一种对不可比量进行加、减、乘、除、取 [根] 的新方法，也就是说，这些量与其他量相比无限大或无限小。” 128对于莱布尼茨来说，算法还不是任何一种明确定义的程序的通用术语，而是指五种（他在这里计算的）基本算术运算。

The problem is manifest in Leibniz’s decision to describe his procedure for differentiation as an “algorithm.” This usage appears, among other places, in the 1684 article in which he first introduces the procedure: “From this rule, known as an algorithm, so to speak, of this calculus, which I call differential, all other differential equations may be found by means of a general calculus.”127 Although Leibniz does call his method an algorithm (Algorithmo) here, he qualifies this characterization with “so to speak” (ut ita dicam), suggesting that the usage is not quite standard. From his other writings, it is apparent that he specifically meant algorithm to imply an analogy with arithmetic. In a 1692 journal article written in French, Leibniz describes the calculus as “a new algorithm [algorithme], that is to say a new manner of adding, subtracting, multiplying, dividing, extracting [roots], proper to incomparable quantities, that is to say those that are infinitely large or infinitely small in comparison to others.”128 For Leibniz, algorithm was not yet a general term for any sort of clearly defined procedure, carrying instead a reference to the five (as he counted them here) basic operations of arithmetic.

莱布尼茨使用“算法”一词，推动了该术语含义的扩展，使其从一组特定的计算方法扩展为一种可应用于其他领域的通用程序。这一过程早在16世纪就已开始，当时迈克尔·斯蒂费尔描述了一种用于处理平方根的“算法”。到了莱布尼茨的时代，一些作家已经开始使用“算法”来指代代数程序。莱布尼茨进一步将该术语扩展到微积分。129这种扩展也在牛顿主义者中扎根。1702 年，普及者约翰·哈里斯将牛顿微积分规则称为“流数的算法或算术”。130因此，早在计算机时代之前，算法这个术语就已经开始指代超出原始算法范围的实践；尽管牛顿甜言蜜语，但计算还是侵入了几何学。

By this use of the word algorithm, Leibniz contributed to the expansion of the term’s meaning from a particular set of computational methods to a general style of procedure that could be applied in other areas. This process had already begun in the sixteenth century, when Michael Stifel described an “algorithm” for manipulating square roots. By Leibniz’s time, some writers had begun to use algorithm to refer to algebraic procedures as well, and Leibniz pushed it further by extending the term to calculus.129 This extension took root among Newtonians as well. In 1702, the popularizer John Harris referred to the rules of Newton’s differential calculus as “the Algorithm or Arithmetick of Fluxions.”130 The term algorithm, then, was already, long before the computer age, coming to refer to practices beyond the original algorism; computation was, in spite of Newton’s blandishment, encroaching on geometry.

然而，人们应该质疑，当莱布尼茨将他的方法称为“可以说是算法”时，他究竟想要单列哪种算术。他的微分过程与普通算术的相似性不如与似是而非的算术——也就是说，与 17 世纪的符号代数的相似性。似是而非的算术并不产生数值结果，而是提供符号公式转换的规则，例如B ( A + E ) = BA + BE或= E。莱布尼茨的“算法”类似地将其规则表述为方程，例如他的乘法规则= xdv + vdx 。虽然莱布尼茨确实启发了算法使用的扩展，但他的追随者将该术语主要应用于这种代数规则。例如，十八世纪后期，德国数学家阿贝尔·布尔哈（Abel Bürja）描述了一种“对数算法”，意为“一种用代数方式表示对数的方法，以及由此产生的计算形式”——即诸如 log a – log b = log 之类的代数规则。131重点不在于这种做法像现代算法一样遵循严格定义的步骤顺序，而在于它涉及用符号表达的运算；因此，“算法”范围的扩展与新符号的发展齐头并进。

One ought to question, however, just what sort of arithmetic Leibniz meant to single out when he called his method an “algorithm, so to speak.” His differentiation procedure is less akin to common arithmetic than to specious arithmetic—that is, to the seventeenth-century version of symbolic algebra. Rather than producing numerical results, specious arithmetic provides rules for the transformation of symbolic formulae, such as B(A + E) = BA + BE or = E. Leibniz’s “algorithm” similarly formulates its rules as equations, as in his multiplication rule = xdv + vdx. While Leibniz did indeed inspire an expansion in the use of algorithm, his followers applied the term mainly to rules of this algebraic sort. For instance, in the late eighteenth century, the German mathematician Abel Bürja described an “algorithm of logarithms,” meaning “a manner of representing them algebraically, with the forms of calculation that result from it”—that is, algebraic rules such as log a – log b = log .131 The point is not that the practice follows a strictly defined sequence of steps, as in modern algorithms, but that it involves operations expressed in symbols; extending the reach of “algorithms” thus went hand in hand with the development of new notations.

在莱布尼茨 1684 年的文章发表后的几十年里，代数已经面临理论上的困难，而微积分使问题变得更加严重。随着算法含义的每次扩展——从算术到代数再到微积分——它所表示的实践的概念合理性变得更具争议性。算术或多或少是没有争议的，关于代数中负数和虚数的疑虑只是偶尔浮现；但关于微积分基础的问题开始充斥着讨论。这场争论的激烈程度部分源于无穷小量的反直觉性质，但也受到对定义符号重要性的认识不断提高的刺激。如下一节所示，莱布尼茨在与哲学家约翰·洛克的交流中被迫为他对符号的使用进行辩护。洛克的工作预示着一批新的认识论的出现，对于这些认识论，符号并不能像对奥特雷德和莱布尼茨那样为语言的不确定性提供解决方案；从洛克的角度来看，符号只是一种语言，其含义与文字一样值得质疑。

Algebra was already facing theoretical difficulties in the decades following Leibniz’s 1684 article, and the calculus made the problem worse. With each expansion of the meaning of algorithm—from arithmetic to algebra to calculus—the conceptual soundness of the practices it denoted became more debatable. Arithmetic was more or less uncontroversial, and the qualms about negative and imaginary numbers in algebra only surfaced now and then; but questions about the foundation of calculus came to pervade the discourse. The intensity of this debate stemmed in part from the counterintuitive nature of infinitesimals, but it was also stoked by an increased awareness of the importance of defining symbols. As the next section shows, Leibniz was forced to defend his use of symbols in his engagement with the philosopher John Locke. Locke’s work heralded a new cluster of epistemologies for which symbols did not offer a solution to the uncertainty of language, as they had for Oughtred and Leibniz; from a Lockean perspective, the symbols simply were a language, and their meanings were just as open to questioning as those of words.

从艺术到自然

From Art to Nature

如果莱布尼茨与牛顿的争论部分是出于个人因素，那么他与洛克的争论则部分地受到了个人因素的缓和。洛克在其 1689 年的《人类理解论》中提出了著名的论断：一切知识都来自感官观察，而不是来自天生的原则或观念。在 1695 年阅读了这篇论文后，莱布尼茨多次试图与洛克辩论，但他无法说服这位英国哲学家认真对待他。从 1700 年开始，莱布尼茨开始写一本书来回应洛克，这本书最终成为《人类理解新论》。洛克于 1704 年去世，出于对死者的尊重，莱布尼茨放弃了他的计划，手稿几十年都未出版。当它最终在 1765 年出版时，洛克已经占了上风。洛克的论证对早期现代普遍和谐的信仰造成了毁灭性的打击，而莱布尼茨是该信仰的后期反对者，这一转变在确定十八世纪符号方法的运作方式方面发挥了重要作用。

If Leibniz’s dispute with Newton was partially motivated by personal factors, his dispute with Locke was partially mitigated by them. In his 1689 Essay Concerning Human Understanding, Locke famously asserted that all knowledge derives from sensory observations rather than from innate principles or ideas. After reading the Essay in 1695, Leibniz repeatedly attempted to engage Locke in a debate, but he could not persuade the English philosopher to take him seriously. Starting in 1700, Leibniz began writing a book-length response to Locke that would eventually become New Essays on Human Understanding. Locke’s death in 1704 led Leibniz to scrap his plans out of respect for the dead, and the manuscript went unpublished for decades. By the time it finally appeared in print in 1765, Locke had already gained the upper hand. Locke’s argument dealt a crushing blow to the early modern belief in universal harmony of which Leibniz was a late holdout, a turn that would play a large role in setting the agenda for the way symbolic methods would function in the eighteenth century.

莱布尼茨对天赋论的辩护的性质和价值一直是学者们争论不休的话题。这场争论在一定程度上与宗教有关；人们普遍认为洛克同情一个被称为索西尼派的异议教派，而莱布尼茨强烈反对该教派。132莱布尼茨的天赋论也被比作伊曼纽尔·康德在十八世纪后期提出的论证，即某些范畴对于所有理性思维都是必需的。133但莱布尼茨的天赋观念领域比康德的先验领域更为广泛。莱布尼茨在 1680 年左右写道，儿童能够轻松理解“数学、道德、法学和形而上学问题”，证明“所有这些概念的种子早已在儿童心中”。134这份学科清单恰好涵盖了莱布尼茨最热切希望应用推理演算的领域，这并非偶然。洛克的论证否定了所有理性思维中都可能存在的前语言“概念”，从而削弱了沃德、威尔金斯以及（我所主张的）莱布尼茨对符号可以规避词语不确定性这一观点的论证方式。因此，普遍特征的可能性——甚至符号方法在数学中的地位——岌岌可危。

The nature and merits of Leibniz’s defense of innatism have been a subject of much debate among scholars. The dispute had to do in part with religion; Locke was widely perceived to sympathize with a dissenting sect known as the Socinians, to whom Leibniz was strongly opposed.132 Leibniz’s innatism has also been likened to Immanuel Kant’s argument, later in the eighteenth century, that certain categories are necessary to all rational thought.133 But Leibniz’s domain of innate ideas was rather more populous than Kant’s transcendental realm. Leibniz wrote some time around 1680 that the ease with which children can understand “mathematics, morals, jurisprudence, and metaphysical matters” proves that “the seeds of all these concepts were already in the child.”134 This list of disciplines, not incidentally, includes just those areas in which Leibniz most fervently hoped to apply the calculus ratiocinator. By denying the existence of prelinguistic “notions” that may be presumed to exist in all rational minds, Locke’s arguments undermined the way Ward, Wilkins, and (I am arguing) Leibniz had justified the view that symbols could circumvent the uncertainty of words. At stake was thus the possibility of the universal characteristic—and perhaps even the status of symbolic methods in mathematics.

在1689年的《论语》中，洛克提出了一个著名的问题：伦理学能否像数学一样确定，并在其中阐述了他眼中的符号计算是如何运作的。在洛克看来，“计算一个长加法、乘法或除法”的过程，只不过是“一种思维的进步，从其自身的角度来看待它”。自己的想法，并考虑它们的一致或不一致。” 135符号起着重要作用，因为“如果不用已知其确切含义的标记和在记忆消失后仍能持久可见的标记来记录各个部分，几乎不可能记住这么多不同的想法。” 136然而，这些符号仅仅是记忆辅助工具，其作用取决于赋予它们的含义。这种对数学符号功能的解释与莱布尼茨在给 Tschirnhaus 的信中的观点完全相反，莱布尼茨认为字符可以引导我们“进入事物的内部”。在莱布尼茨看来，伦理论证的不确定性主要源于我们用来表达它们的字符不像数学符号那样“排列”得很好。相反，对洛克来说，仅仅改进符号不足以使伦理学像数学一样具有论证性。问题在于观念的形成，而观念不能仅靠字符来“固定”。

In the 1689 Essay, Locke famously considered whether ethics could be made as certain as mathematics, in the course of which he gives an account of how, in his view, symbolic computation works. In Locke’s view, the process of “casting up a long sum, either in addition, multiplication, or division” is nothing but “a progression of the mind, taking a view of its own ideas, and considering their agreement or disagreement.”135 Notation plays an important role, since “without setting down the several parts by marks, whose precise significations are known, and by marks, that last and remain in view, when the memory had let them go, it would be almost impossible to carry so many different ideas in mind.”136 Yet these symbols are mere memory aids that are only as good as the meanings assigned to them. This account of the function of mathematical symbols goes directly against Leibniz’s case, in the letter to Tschirnhaus, that characters can lead us “into the interior of things.” In Leibniz’s view, the uncertainty of ethical arguments stems primarily from the fact that the characters we use to express them are not as well “arranged” as mathematical notations. For Locke, by contrast, simply improving the notation would not suffice to make ethics demonstrative like mathematics. The problem lies in the formation of the ideas, which cannot be “fixed” by means of characters alone.

洛克对符号认知功能的论述揭示了一个现实特征观倾向于掩盖的难题。洛克认为，词语并非指代事物本身，而仅指他所谓的名义本质——任意构建的性质集合，例如光滑的红球。名义主义在经院哲学中由来已久，但洛克与早期名义主义者的不同之处在于，他认为语言差异是不可约化的。洛克指出，词语是“自愿的符号”，这意味着每个人都有“自由，可以用词语代表他喜欢的思想”。137然而，人们无权控制他人如何理解词语。因此，假设人们可以绕过意义对说话者和听话者头脑中是否存在可通约思想的依赖——假设词语可以指代他人头脑中的想法或事物本身，而无需经过心理概念的中介——就是将语言的功能推向了其合法范围之外。这种关于“词语滥用”的论述，构成了洛克对培根及其追随者长期以来所哀叹的语言混乱的解释。138洛克认为，问题不仅仅在于人们目前所使用的语言的腐败本质。问题在于，语言本身，无论以口头还是书面形式表达，都从根本上无法建立培根学派所期望的那种透明的沟通。

Locke’s account of the cognitive function of signs surfaced a difficulty that the real-character view tended to gloss over. Words, Locke argues, refer not to things themselves but only to what he called nominal essences—arbitrarily constructed bundles of qualities such as smooth red sphere. Nominalism had a long history in Scholastic thought, but Locke differs from earlier nominalists in the extent to which he treats linguistic differences as irreducible. Locke states that words are “voluntary signs,” meaning that everyone has “a liberty, to make words stand for what ideas he pleases.”137 One does not, however, have the liberty to control how other people understand words. As a result, to suppose that one can get around the dependence of signification on the existence of commensurable ideas in the minds of the speaker and hearer—to act as if words can refer either to ideas in other people’s heads or to things themselves, unmediated by mental conceptions—is to push language beyond what it can legitimately do. This account of the “abuse of words” constitutes Locke’s explanation for the linguistic confusion that Bacon and his followers had long lamented.138 The problem, as Locke has it, is not just the corrupt nature of the languages people speak at present. The problem is that language itself, meaning any form of signification whether spoken or written, is fundamentally incapable of establishing the sort of transparent communication that the Baconians desired.

莱布尼茨或许并非洛克在这场论证中的直接目标，但威尔金斯或许是。正如汉娜·道森所强调的，洛克坚持符号是“自发的”，这与威尔金斯等17世纪符号学家对概念无可争议性的信心背道而驰。139洛克论证这一观点的部分方式是列举一些思维方式与他自身截然不同的文化。国家。例如，他列举了各种各样的社会，这些社会据称将婴儿暴露在荒野中，杀死达到一定年龄的人，并将自己的孩子养肥并吃掉。140洛克从不可靠的游记中借用的这些原始人类学概述，旨在证明道德观念在不同文化中的发展非常不同，因此不可能是在出生时就被神圣地铭刻在头脑中的。洛克不是道德相对主义者——尽管他有异端邪说，但他是一位虔诚的基督徒——但他的哲学确实对一个真实的人物可以在道德问题上达成普遍共识的想法产生了可怕的影响。对洛克来说，沟通的可能性取决于对符号含义的一致看法，而简单地在表格中排列符号并不能保证这种一致。

Leibniz was probably not one of Locke’s immediate targets in this argument, but Wilkins probably was. As Hannah Dawson has stressed, Locke’s insistence that signs are “voluntary” was directed against the confidence of seventeenth-century semioticians like Wilkins in the unproblematic nature of concepts.139 Locke argues this point in part by giving examples of cultures whose ways of thinking differ radically from those of his own country. For instance, he lists a variety of societies that supposedly exposed infants in the wilderness, killed people once they reached certain ages, and fatted and ate their own children.140 These protoanthropological sketches, which Locke borrowed from unreliable travel narratives, are supposed to prove that moral ideas develop very differently across cultures and thus cannot have been divinely etched on the mind at birth. Locke was no moral relativist—he was, for all his heterodoxy, an ardent Christian—but his philosophy did have dire implications for the idea that a real character could establish universal agreement on moral issues. The possibility of communication, for Locke, depends on the existence of an accord about what signs mean, and simply arranging symbols in a table can provide no guarantee of such an accord.

当洛克发表这一论证时，威尔金斯的计划早已不堪重负而告失败，但莱布尼茨的普适性理论依然鲜活，其提出者在《新论》中着手指出洛克低估了符号在推理中的作用。与洛克相反，莱布尼茨认为数学思想是与生俱来的，自出生起就存在于头脑中，等待着被揭示。他引用尤利乌斯·斯卡利杰的话，将这些与生俱来的思想描述为“隐藏在我们内心深处的鲜活的火焰或闪光，但通过感官刺激就能显现出来，就像钢铁碰撞出火花一样。” 141人们或许会质疑这场争论是否只是小题大做——思想是通过感官输入进入头脑，还是仅仅被感官“激发”而来，这真的重要吗？但答案与符号方法息息相关。在洛克的理性活动概念中，盲目的思考是没有立足之地的；人们必须将清晰形成的思想组合起来，并将其与符号联系起来，否则它们就只是无稽之谈。莱布尼茨的天赋使他能够将符号视为揭示其概念基础尚未被发现的新真理的手段。

Wilkins’s project had already collapsed under its own weight by the time Locke published this argument, but Leibniz’s universal characteristic was still alive, and its projector set out to show in New Essays that Locke was undervaluing the role of signs in reasoning. In opposition to Locke, Leibniz maintained that mathematical ideas were innate, present in the mind from birth and waiting to be revealed. Citing Julius Scaliger, he describes these innate ideas as “living fires or flashes of light hidden inside us but made visible by the stimulation of the senses, as sparks can be struck from a steel.”141 One might question whether this dispute was much ado about nothing—does it really matter whether ideas come into the mind through sensory input or are merely “sparked” by it? But the answer makes a difference in regard to symbolic methods. Blind thought had no place in Locke’s conception of rational activity; one had to assemble clearly formed ideas to attach to the symbols or they would amount to mere nonsense. Leibniz’s innatism enabled him to treat symbols, instead, as a means of revealing new truths whose conceptual bases had not yet been discovered.

数学之所以出现在这场辩论中，部分原因是洛克曾用它作为天赋论证难题的一个例子。洛克认为，如果人们接受心灵对数学原理具有“内隐知识”，那么“所有数学论证，以及基本原理，都必须被视为心灵中与生俱来的印象：我担心，那些发现论证命题比接受论证更难的人，很少会承认这一点。” 142也就是说，如果发现数学知识如此困难，那么数学知识又怎么可能是与生俱来的呢？莱布尼茨虽然承认数学很难学，但他仍然反驳说，数学是建立在与生俱来的思想之上的：“认为我们可以轻松地在灵魂中解读这些永恒的理性法则，就像可以在布告栏上读到法官的法令一样，无需努力或探究，这确实是错误的；但只要它们能够被发现就足够了。”通过注意力，我们可以发现内在的奥秘：感官提供了机会，成功的实验也有助于佐证推理，就像算术检验可以帮助我们避免在冗长的推理过程中出现计算错误一样。” 143莱布尼茨在这里赋予感官的角色，很像他在《发现的艺术》中所描述的普遍特性：激发思考，检验结论。感官并非我们所有知识的来源，但它们提供了一种必要的手段来“纠正”我们的推理，以便我们能够更好地发现我们内在的东西。

Mathematics arose in this debate in part because Locke had used it as an example of the difficulties surrounding innatism. If one accepts that the mind has an “implicit knowledge” of mathematical principles, Locke had argued, then “all mathematical demonstrations, as well as first principles, must be received as native impressions on the mind: which I fear they will scarce allow them to be, who find it harder to demonstrate a proposition, than assent to it, when demonstrated.”142 How, that is, can mathematical knowledge be innate if discovering it can be arduously difficult? While Leibniz concedes that mathematics can be hard to learn, he nonetheless counters that it is built from ideas that exist in the mind from birth: “It would indeed be wrong to think that we can easily read these eternal laws of reason in the soul, as the Praetor’s edict can be read on his notice-board, without effort or inquiry; but it is enough that they can be discovered within us by dint of attention: the senses provide the occasion, and successful experiments also serve to corroborate reason, somewhat as checks in arithmetic help us to avoid errors of calculation in long chains of reasoning.”143 The role Leibniz here assigns to the senses is much like that of the universal characteristic as he describes it in “The Art of Discovery”: sparking thought and testing conclusions. The senses are not the source of all our knowledge, but they provide a necessary means of “rectifying” our reasoning so that we can better discover what is within us.

这两种立场意味着对符号在思想中的作用持不同态度。洛克认为，符号在能指和所指方面都是任意的：人们可以自由选择如何称呼树木以及如何定义树木。因此，符号就像词语一样，并不比定义它们的理念更好。相比之下，莱布尼茨认为，我们用来记录理念的“可感知痕迹”在发现理念本身的过程中起着至关重要的作用：“我们不可能拥有不需要任何可感知事物的抽象思维，即使这种抽象思维仅仅是诸如字母形状或声音之类的符号；尽管这些任意的符号与这些思维之间没有必然联系。” 144莱布尼茨继续说道，如果感官输入对思维来说不是必需的，“身体与灵魂之间预先设定的和谐”就不会存在。145这种思路表明，符号可以揭示事物真实本质的某些信息，因为它们参与了灵魂与世界之间神圣设立的和谐。这也表明，从更世俗的层面来看，沟通问题并不像洛克声称的那样棘手。即使符号本身是任意的，它们所指代的观念却并非如此，因此，确保它们对每个人的意义相同并不需要付出巨大的努力。

These two positions entail different attitudes toward the role of signs in thought. For Locke, signs are arbitrary in both the signifier and the signified: one is free to choose both what to call trees and how to define a tree. Symbols are thus, just like words, no better than the ideas by which they are defined. For Leibniz, by contrast, the “sensible traces” we use to record our ideas play an essential role in the discovery of the ideas themselves: “we cannot have abstract thoughts which have no need of something sensible, even if it be merely symbols such as the shapes of letters, or sounds; though there is no necessary connection between such arbitrary symbols and such thoughts.”144 If sensory input were not necessary for thought, Leibniz continues, “the pre-established harmony between body and soul” would not obtain.145 This line of thought suggests that symbols can reveal something about the true natures of things, since they take part in the divinely instituted harmony between the soul and the world. It also suggests, at a more mundane level, that the problem of communication is less intractable than Locke claimed. Even if the symbols themselves are arbitrary, the ideas to which they refer are not, and as a result it would not take a Herculean effort to ensure that they mean the same thing to everyone.

这种使用符号的理论基础，最终取决于早期现代信仰的一种特殊变体，即人类思维是世界的一面镜子。在莱布尼茨的宇宙中，每个存在体都能感知其他存在体；我们之所以无法知晓一切，唯一的原因是这些感知中的大多数在某种程度上是混乱的。莱布尼茨在《新散文》的前言中写道：“甚至可以说，凭借这些细微的感知，现在与未来融为一体，背负着过去，万物和谐共存……即使拥有上帝般敏锐的眼睛，也能从最卑微的物质中读出宇宙事件的整个序列。” 146和谐学说使莱布尼茨能够认识到自然语言的历史偶然性，同时又不放弃这样的信念：符号——无论是真实的字符还是声音——能够以不容置疑的方式传达数学、道德、形而上学和其他领域的真理。如果洛克指出一种文化相对主义，承认道德观念依赖于特定国家的习俗，那么莱布尼茨采取了相反的姿态：通过与思想相协调而不是按照惯例来表示思想，普遍特征会使文化短路。

This rationale for the use of symbols depends, ultimately, on an idiosyncratic variant of the early modern belief that the human mind is a mirror of the world. In a Leibnizian universe, every being perceives every other being; the only reason we do not know everything is that most of these perceptions are to some degree confused. “It can even be said,” Leibniz writes in the preface to New Essays, “that by virtue of these minute perceptions the present is big with the future and burdened with the past, that all things harmonize . . . and that eyes as piercing as God’s could read in the lowliest substance the universe’s whole sequence of events.”146 The doctrine of harmony enabled Leibniz to recognize the historical contingency of natural languages without giving up on the belief that symbols—be they real characters or sounds—could convey truths about mathematics, morals, metaphysics, and other domains in ways that left no room for disagreement. If Locke pointed toward a form of cultural relativism that recognized the dependence of moral ideas on the customs of particular countries, Leibniz gestured the opposite way: by harmonizing with ideas rather than signifying them by convention, the universal characteristic would short-circuit culture.

到 1765 年莱布尼茨的《人类理解新论》最终出版时，他的对手已经占据了上风。洛克的《人类理解新论》最初被牛津大学斥为异端，并立即遭到多方面的批评。147但洛克思想后来成为 18 世纪占主导地位的哲学流派之一，而莱布尼茨则处于劣势。1755 年里斯本地震——伏尔泰在《老实人》和《里斯本灾难诗》中对此进行了著名的描述——只是导致莱布尼茨乐观形而上学受到质疑的众多因素之一。伏尔泰的诗讽刺了莱布尼茨“一切都是必然的”学说，这意味着在地震中丧生的婴儿和儿童一定是在某种程度上罪有应得。148在不那么现实的层面上，关于语言的观念在十八世纪中叶发生了变化，以至于字符和语言之间的区别逐渐消失。威尔金斯的真正性格未能被人们接受，从17世纪80年代开始，人们的观点开始反对这样的观点：无论符号如何“人为”地设计，都不足以确保人们正确理解事物；人们越来越认识到，要疏通沟通的“管道”，仅仅依靠一种新的书写形式是不够的。

By the time Leibniz’s New Essays on Human Understanding was finally published in 1765, his opponent had secured the high ground. Locke’s Essay was initially condemned by Oxford for heterodoxy, and it faced immediate criticism from numerous quarters.147 But Lockean thought would go on to become one of the dominant philosophical strains in the eighteenth century, whereas the tide turned against Leibniz. The 1755 Lisbon earthquake, which Voltaire famously depicted in Candide and in his “Poem on the Lisbon Disaster,” was only one of a number of factors that placed Leibniz’s optimistic metaphysics under question. Voltaire’s poem satirizes Leibniz’s doctrine that “all is necessary,” which would have entailed that the infants and children who died in the earthquake must have in some way deserved their fate.148 At a less existential level, ideas about language had changed by the mid-eighteenth century so as to erode the distinction between characters and languages. Wilkins’s real character had failed to catch on, and starting in the 1680s, opinion turned against the idea that symbols, no matter how “artificially” designed, could suffice to ensure that people were understanding things correctly; clearing up the “pipes” of communication, it was increasingly recognized, would take more than a new form of writing.

洛克的著作为这些浮夸的方案提供了一个极具吸引力且谦逊的替代方案。洛克并不依赖符号的力量来克服文化差异，而是强调定义——这是一种通过阐明思想的形成方式来解决局部概念问题的务实方法。他还鼓励培养良好的习惯，使理性成为第二天性。在《教育思考》（169​​3年）中，他反对在教学中使用明确的规则：“不应使用那些总是会从孩子们记忆中消失的规则来教导他们。你认为他们必须做的事情，只要有机会，就通过必不可少的练习，让他们牢牢记住；如果可能的话，创造机会。这会在他们身上养成习惯，一旦养成，习惯就会自然而然地发挥作用，无需记忆的帮助。” 149洛克在这里谈论的并非计算规则——他指的是像“晚餐前不吃布丁”这样的规则——但他的论证对之后一个多世纪的数学教学产生了重大影响。对洛克来说，明确的指示并非塑造人类行为的有效手段；正如他后来在逻辑学中更多地写道的那样，更好的做法是“养成不去思考规则就去做事的习惯” 。150

Locke’s work offered an appealingly humble alternative to such high-flown schemes. Rather than trusting in the power of symbols to overcome cultural differences, Locke emphasized definitions—a pragmatic measure for addressing local conceptual problems by clarifying how ideas were assembled. He also encouraged cultivating good habits so that rationality would become second nature. In Some Thoughts Concerning Education (1693), he argues against the use of explicit rules in pedagogy: “Children are not to be taught by Rules which will be always slipping out of their Memories. What you think necessary for them to do, settle in them by an indispensible Practice, as often as the Occasion returns; and if it be possible, make Occasions. This will beget Habits in them, which being once establish’d, operate of themselves easily and naturally, without the Assistance of the Memory.”149 Locke is not talking about computational rules here—he means rules like “No pudding before dinner”—but his argument would have significant influence on mathematical pedagogy for well over a century. Explicit directions, for Locke, are an ineffective means of shaping human behavior; it is better, as he later wrote more in the context of logic, to “settle the habit of doing without reflecting on the rule.”150

无论是否受到洛克的直接影响，十八世纪数学家们普遍追随他，强调定义是建立清晰度的一种手段，并避免使用明确的规则，而倾向于理论解释，以使程序直观明显。一些作家，尤其是英国的作家，开始对符号产生怀疑。151约翰·基尔 (John Kiell)在其广泛重印的十八世纪欧几里得《几何原本》版的前言中宣称：“所有科学的要素都应该按照最简单的方法来处理，而不应该涉及符号、注释或晦涩的原理。” 152这种转变也影响了英国微积分的普及，他们（虽然他们大量使用符号）倾向于追随牛顿，强调几何而不是代数。 1742 年，柯林·麦克劳林 (Colin MacLaurin) 在早期牛顿微积分教科书《流数论》中指出，代数“可能被用来用复杂的符号掩盖深奥的学说，这些学说无法用简单的几何形式很好地表达出来。” 153麦克劳林继续说，避免这种晦涩的关键是“清楚地定义符号的含义和用法，然后谨慎行事。” 154这与奥特雷德一个世纪前的说法形成了鲜明的对比。符号不再被视为一种将数学思想直接写在纸上而不受文字干扰的方式；相反，人们怀疑符号是没有意义的，除非证明它们另有含义。

Whether through Locke’s direct influence or not, eighteenth-century mathematicians generally followed him in emphasizing definitions as a means of establishing clarity, as well as in avoiding explicit rules in favor of theoretical explanations that were supposed to make procedures intuitively obvious. Some authors, especially in Britain, grew suspicious of symbols.151 In his preface to a widely reprinted eighteenth-century edition of Euclid’s Elements, John Kiell declares, “The Elements of all Sciences ought to be handled after the most simple Method, and not to be involved in Symbols, Notes, or obscure Principles.”152 The turn also affected British popularizations of calculus, which (while they made liberal use of symbols) tended to follow Newton in emphasizing geometry over algebra. In his 1742 Treatise of Fluxions, an early Newtonian calculus textbook, Colin MacLaurin states that algebra “may have been employed to cover, under a complication of symbols, abstruse doctrines, that could not bear the light so well in a plain geometrical form.”153 The key to avoiding this obscurity, MacLaurin continues, is “defining clearly the import and use of the symbols, and proceeding with care afterwards.”154 The contrast with Oughtred’s statement of a century before is stark. Symbols are no longer treated as a way of placing mathematical ideas directly on the page without the interference of words; instead, they are suspected to be meaningless until proven otherwise.

大陆数学家对莱布尼茨的同情远胜于麦克劳林，他们的清晰度概念更多地受到笛卡尔的影响。符号方法的发展有增无减。然而，法国和德国传统也更加注重澄清符号的含义。让·勒朗·达朗贝尔为法国启蒙运动的伟大纪念碑《百科全书》撰写了大部分数学条目，他非常注意定义诸如负数和微分等棘手的概念。例如，他将微分（différentiel）定义为“无限小或小于所有可指定量的量” 。155他在关于负数（négatif ）的文章中首先指出，一些数学家将负数定义为“小于零”的数，但他认为这个定义是错误的；他详细地解释了负数，尽管有些棘手，但必须将其理解为与某些其他数成对立关系的正数，就像两条曲线可以从给定点向相反方向延伸一样。156与麦克劳林一样，达朗贝尔即使在使用符号表示法时也赋予几何可解释性以特权。达朗贝尔的数学知识概念不接受没有概念基础支撑的方法，无论这种基础多么抽象。

Continental mathematicians were more sympathetic to Leibniz than MacLaurin was, and their ideas of clarity were more under the sway of Descartes. The development of symbolic methods continued unabated. Yet the French and German traditions also increased their focus on clarifying the meanings of symbols. Jean Le Rond d’Alembert, who wrote most of the entries on mathematics for the great monument of the French Enlightenment, the Encyclopédie, paid great attention to defining such tricky concepts as negative numbers and differentials. He defines differential (différentiel), for instance, as “a quantity infinitely small, or less than all assignable magnitude.”155 His article on negative (négatif) begins by noting that some mathematicians define negatives as numbers “smaller than zero,” but he thinks this definition is wrong; negatives, he explains in great if somewhat knotty detail, must be understood as positive numbers placed in a relation of opposition to certain other numbers, as two curves may extend in opposite directions from a given point.156 Like MacLaurin, d’Alembert gives a privileged role to geometric interpretability even when employing symbolic notation. D’Alembert’s conception of mathematical knowledge had no place for methods that were not backed up with some conceptual basis, abstract as that basis may be.

这种思想转变并没有彻底摧毁莱布尼茨的声誉。将“计算”扩展为通用方法的想法，在19世纪查尔斯·巴贝奇和艾达·洛夫莱斯的著作中获得了新的生命。威尔金斯的真实性格也是如此：在回忆录中，巴贝奇讲述了他年轻时试图创建一个类似的系统。157莱布尼茨的逻辑著作在十九世纪后期重新受到关注，因为布尔、弗雷格和罗素在过去寻找志同道合的人。因此，一些计算史直接从莱布尼茨延续到十九世纪巴贝奇、布尔或朱塞佩·皮亚诺的作品。158但跳到这次复兴就会错过一些重要的东西。正如下一章所示，在莱布尼茨和巴贝奇之间的一个世纪里，通用计算的想法并没有消失；它只是采取了不同的形式。十八世纪通用计算版本的顶峰出现在尼古拉·德·孔多塞的作品中，他勾勒出了一种类似于推理演算的东西，其认识论更符合洛克的观点，而不是莱布尼茨的观点。对于孔多塞来说，该系统的目的不是揭示所有聪明人心中已经存在的想法；而是用更好的世界观取代人们现有的世界观。

This intellectual shift did not defeat Leibniz’s reputation for good. The idea of extending “calculation” into a universal method would gain a new life in the nineteenth-century work of Charles Babbage and Ada Lovelace, as would Wilkins’s real character: in his memoir, Babbage recounts a youthful attempt to create a comparable system.157 Leibniz’s writings on logic would return to prominence later in the nineteenth century as Boole, Frege, and Russell looked for a kindred spirit in the past. As a result, some histories of computation proceed directly from Leibniz to the nineteenth-century work of Babbage, Boole, or Giuseppe Peano.158 But to skip ahead to this revival would be to miss something important. As the next chapter shows, the idea of universal computation was not absent in the century between Leibniz and Babbage; it simply took on a different form. The culmination of this eighteenth-century version of universal computation appeared in the work of Nicolas de Condorcet, who sketched out something like a calculus ratiocinator within an epistemology more in line with Locke’s views than with Leibniz’s. For Condorcet, the purpose of the system was not to reveal ideas that were already present in the minds of all intelligent people; it was to replace people’s existing conceptions of the world with better ones.

分裂

The Schism

1794 年冬，孔多塞侯爵玛丽让安托万尼古拉卡里塔知道自己活不了多久了。孔多塞最为人熟知的身份是数学家和政治思想家，他也是 1789 年法国大革命的积极参与者。作为旧制度下的贵族，他在 1791 年直言不讳地宣布支持共和国，令在场众人震惊。1然而，国民议会解散仅一年多后，他对温和派吉伦特派的支持却使他沦为逃犯。1793 年 10 月 3 日，雅各宾派发布了对他的逮捕令，他被迫离家出走。2在八个月的躲藏期间，他靠撰写政治文章来打发时间，但这些文章丝毫没有反映出他处境的悲惨。在这​​些手稿中，包括他的一些最著名的作品，他勾勒出了一个与法国大革命的现实形成鲜明对比的人类乌托邦计划。

In the winter of 1794, Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet, knew that he would not live long. Best known as a mathematician and political thinker, Condorcet was an enthusiastic participant in the French Revolution of 1789. A titled aristocrat in the Old Regime, he shocked an audience in 1791 with a forthright declaration of support for the republic.1 Just over a year after the National Assembly was dissolved, however, his support for the moderate Girondin faction made him into an outlaw. On October 3, 1793, the Jacobins released a warrant for his arrest, and he was forced to flee from his home.2 During his eight months in hiding, he passed the time by writing political texts that show little hint of the direness of his situation. In these manuscripts, which include some of his best-known work, he sketched out a utopian plan for humanity that stood in sharp contrast to the realities of revolutionary France.

孔多塞在逃亡期间写下的一篇残篇，后来以《人类精神进步历史图景素描》为名出版，其中提出了两种确保人类进步的方法：第一，采用“技术方法”，他指的是“将大量主题排列成一个系统，以便我们能够直接掌握它们之间的关系，快速感知它们的组合，并很容易地从中形成新的组合”；第二，采用“通用语言”，“用符号表达真实的对象本身，或明确定义的科学语言是由简单而一般的观念组成的集合，这些观念在所有人心中都是相同的，或者可能以类似的形式出现，或者这些观念之间的一般关系，人类思维的运作，或个别科学或艺术程序所特有的运作。” 3他写道，这样的语言不会有“与白话不同的科学习语的缺点”；它可以被所有人学习，就像小学生学习代数语言一样，提供普遍获取最佳知识的途径，并确保对术语的含义或论证的有效性不会产生分歧。4

In one of the fragments he wrote while a fugitive, later published as Sketch for a Historical Picture of the Progress of the Human Mind, Condorcet suggests two means by which the improvement of the human race can be assured: first, the adoption of “technical methods,” by which he means “the art of arranging a large number of subjects in a system so that we may straightaway grasp their relations, quickly perceive their combinations and readily form new combinations out of them”; and second, a “universal language” that “expresses by signs either real objects themselves, or well-defined collections composed of simple and general ideas, which are found to be the same or may arise in a similar form in the minds of all men, or the general relations holding between these ideas, the operations of the human mind, or the operations peculiar to the individual sciences or the procedures of the arts.”3 Such a language, he writes, would not have “the disadvantages of a scientific idiom different from the vernacular”; it could be learned by all, as schoolchildren learn the language of algebra, providing universal access to the best knowledge available and ensuring that there could be no disagreement about either the meaning of terms or the validity of arguments.4

孔多塞的著作长期以来一直是将数学推理扩展到所有知识领域的愿望的转喻。在《事物的秩序》中，米歇尔·福柯称孔多塞是试图在物理学和天文学以外的领域“将经验知识数学化”的法国古典思想家之一。5同样，翁贝托·埃科在《寻找完美的语言》中，以孔多塞创造通用语言的方法作为“对完美语言的追求最终转向逻辑数学演算”的证据，罗杰·查蒂尔将孔多塞的项目作为渴望“形式化认知操作和逻辑推理”的一个例子。6事实上，孔多塞的传记作者基思·迈克尔·贝克认为，他的一些言论预见了二十世纪计算领域的发展。7但孔多塞对待符号方法的交流方面的方式与后来的逻辑和计算实践者截然不同。与现代编程语言不同，孔多塞的系统不仅旨在产生结果，还旨在提供一套表格，对世间万物进行分类。因此，孔多塞的项目体现了一种独特的十八世纪象征主义方法，即使推理变成了一个机械过程，它仍然以人类思维为中心。

Condorcet’s work has long stood as a metonym for the desire to extend mathematical reasoning to all domains of knowledge. In The Order of Things, Michel Foucault cites Condorcet as one of the French classical thinkers who attempted “to mathematize empirical knowledge” in domains outside of physics and astronomy.5 Similarly, Umberto Eco, in The Search for the Perfect Language, takes Condorcet’s approach to creating a universal language as proof that “the search for perfect languages was definitively turning in the direction of a logico-mathematical calculus,” and Roger Chartier holds Condorcet’s project up as an example of a desire for “formalizing cognitive operations and logical reasoning.”6 Indeed, Condorcet’s biographer Keith Michael Baker argues that some of his remarks anticipate twentieth-century developments in computation.7 But Condorcet treated the communicational side of symbolic methods very differently from later practitioners of logic and computation. Unlike a modern programming language, Condorcet’s system was meant not just to produce results but also to provide a set of tables that classify all the things in the world. Condorcet’s project thus exemplifies a distinctly eighteenth-century approach to symbolism, one that keeps the human mind at the center even as reasoning becomes a mechanical process.

孔多塞未能亲眼见证他关于通用语言的评论得以发表。1794 年 3 月 27 日，他在试图逃离藏身之所时被捕，两天后，他在牢房中不明原因去世。然而，他的论文幸存了下来，《代数概要》于次年出版。除了《代数概要》的手稿外，他还留下了一份未完成的通用语言计划。8在约 90 页的手写文件中，他展示了如何使代数符号语言独立存在，而无需用句子将各个部分连接在一起（图 3.1）。该方法基于方框符号，指示哪些方程被视为给定方程、哪些方程被视为假设方程、哪些方程已被证明方程等等；他还引入了表示不同类型数字以及函数、级数、近似值和条件表达式。虽然他从代数入手，但他的目标是将这种符号化扩展到所有科学领域。手稿包含几何、力学、天文学和自然哲学的基本符号，以及一种以象形文字形式表达某些思想的“象形文字”语言的提案。9在手稿结束之前，孔多塞承诺解释他的体系如何扩展到形而上学、语言学、道德学和政治学。10

Condorcet did not live to see his remarks about the universal language published. On March 27, 1794, he was arrested while attempting to flee the house where he was hiding, and two days later he died in his cell of unknown causes. His papers, however, survived, and the Sketch was published the following year. Along with the manuscript for the Sketch, he left behind an unfinished plan for the universal language.8 Over the course of about ninety handwritten pages, he shows how the symbolic language of algebra can be made to subsist on its own, without the need for sentences linking the pieces together (figure 3.1). The method is based on boxlike symbols indicating which equations are taken as given, supposed as hypotheses, proven, and so forth; he also introduces symbols for different types of number as well as for functions, series, approximations, and conditional expressions. Although he begins with algebra, his goal is to extend this symbolization to all sciences. The manuscript includes rudimentary notations for geometry, mechanics, astronomy, and natural philosophy as well as a proposal for a “hieroglyphic” language that represents certain ideas pictographically.9 Just before the manuscript cuts off, Condorcet promises to explain how his system can be extended to metaphysics, linguistics, morals, and politics.10

这个项目出现于 17 世纪人们对符号抱有的乌托邦式希望广泛复兴的背景下。在 1785 年的一封信中，孔多塞引用了 W. W. 莱布尼茨作为这一思想的先例，尽管他可能不知道莱布尼茨试图赋予符号普遍性的细节。11作为另一个潜在来源，2004 年版孔多塞手稿的编辑们建议使用皮埃尔·埃里戈内的《数学证明》 ——这本 17 世纪早期的书通过设计为可用拉丁语和法语阅读的符号来呈现数学证明。12孔多塞是 18 世纪最后十年重新审视此类思想的众多思想家之一。几年后，在 1797 年，约瑟夫·德·迈米厄出版了他所谓的《符号表》：一套被认为具有数学符号跨语言特性的符号系统。13然而，孔多塞追求的不仅仅是国际交流。他还希望通过发展一种即使所有现存语言都被遗忘，子孙后代也能阅读的书写形式，来保存知识，避免灾难。

This project appeared amid a broad revival of the utopian hopes that had attended symbols in the seventeenth century. In a 1785 letter, Condorcet had cited G. W. Leibniz as a precedent for the idea, although he probably did not know the details of Leibniz’s attempts at a universal characteristic.11 As another potential source, the editors of the 2004 edition of Condorcet’s manuscript suggest Pierre Hérigone’s Cvrsvs mathematicvs—the early-seventeenth-century book that presents mathematical proofs through symbols designed to be readable in both Latin and French.12 Condorcet was one of a number of thinkers who revisited such ideas in the last decade of the eighteenth century. A few years later, in 1797, Joseph de Maimieux published what he called the Pasigraphie: a system of symbols that were supposed to share the translinguistic character of mathematical notations.13 Condorcet, however, was after more than just international communication. He also wanted to preserve knowledge against catastrophe by developing a form of writing that future generations could read even if all existing languages were forgotten.

鉴于孔多塞的设想，不难理解为何他的系统会受到一些观察家的关注，并被认为是原始计算的。他的通用代数暗示了编程语言的许多元素，包括循环、if-then 条件语句以及变量绑定。例如，使用划掉的字母NI表示整数，他写下以下内容来表示如果n为偶数，则值为 + a ，否则为-a：14

Given what Condorcet came up with, it is not hard to see why his system has struck some observers as protocomputational. His universal algebra hints at many of the elements of programming languages, including loops, if–then conditionals, and the binding of variables. For instance, using the struck-through letters NI to mean integer, he writes the following to indicate the value +a if n is even and −a otherwise:14

这里有很多内容预示了后来的发展；事实上，用m来指代任意整数似乎预示了现代符号逻辑的“存在”量词。但孔多塞的工作指向了与从早期现代代数经莱布尼茨和乔治·布尔到计算机的历史潮流不同的方向。15与莱布尼茨和布尔不同，孔多塞的逻辑符号与代数运算符完全不同，他为几何学和天文学等其他学科赋予了它们自己的符号，包括特殊的线条来指示每个符号所属的科学。（这就是整数符号似乎被划掉的原因。）他没有试图开发一种适用于所有科学的单一方法，而是致力于将多种科学汇集在一起​​；他没有提出如何使用符号的正式规则，而是专注于教人们符号的含义，而将操作留给理性来指导。

There is much here that anticipates later developments; indeed, the use of m to refer to an arbitrary integer seems to anticipate the “there exists” quantifier of modern symbolic logic. But Condorcet’s work points in a different direction from the historical current that leads from early modern algebra, via Leibniz and George Boole, to the computer.15 In contrast to Leibniz and Boole, Condorcet makes his logical notations entirely distinct from algebraic operators, and he gives other sciences such as geometry and astronomy their own symbols, including special lines indicating to which science each symbol belongs. (This is why the symbol for integer appears to be crossed out.) Instead of trying to develop a single method that could apply to all sciences, he is out to gather multiple sciences together; instead of presenting formal rules for how to use the symbols, he focuses on teaching people what the symbols mean, leaving the operations to be guided by reason.

在自然科学中，该系统通过一种排序方法运作，福柯认为这种方法具有启蒙时期的特色：分类法。例如，可以用数字 145702342 表示一种植物，其中 145 代表纲，70 代表属，23 代表种，42 代表变种。16这种数字分类构成了孔多塞的“技术方法”，它既为科学写作提供了一种符号，也提供了一个可与词典或档案室配合使用的查找系统。他不遗余力地展示了如何在不使用任何现有语言的情况下解释符号的含义——例如，通过图片、类比和实验说明。这种语言的再创造以特里斯特拉姆·沃尔夫所说的语言唯意志论为前提——这种观点认为，既然符号的含义是任意的，那么这些含义可以而且应该被改变，以更好地适应事物的本质。17孔多塞解释说，新术语将通过明确的定义而非默契来建立。18正如他在几何学讨论中所述，他的目标是勾勒出一个足够完整的系统，以便新符号可以仅使用已经引入的其他符号来定义，“不受所有先前约定的约束” 。19

In the natural sciences, this system works by the ordering device that is, for Foucault, distinctive of the Enlightenment period: the taxonomy. For example, one could have the number 145702342 designate a plant, with 145 representing the class, 70 the genus, 23 the species, and 42 a particular variety.16 This numerical classification, which constitutes Condorcet’s “technical method,” would provide both a notation for scientific writing and a lookup system that could work with a dictionary or filing bureau. He goes to great length to show how one can explain the meanings of signs without using any preexisting language—for instance, through pictures, analogies, and experimental instructions. This re-creation of language was premised on what Tristram Wolff has called linguistic voluntarism—the idea that, since the meanings of signs are arbitrary, those meanings can and should be altered to better suit the natures of things.17 New terms, Condorcet explains, will be established through explicit definitions, not tacit agreement.18 As he states in the discussion of geometry, his goal is to sketch out a system complete enough that new symbols may be defined using nothing but the other symbols already introduced, “independently of all anterior convention.”19

然而，甚至在孔多塞开始超越数学之前，他就开始对语言重新回归的可能性表现出一些担忧。在通用代数能够应用于特定情况之前，有必要确定符号的含义，而这并非显然，这总是可以做到的，无需借助任何言语。孔多塞认为这是他方案的一个缺陷，尽管并非致命缺陷：

Even before Condorcet begins to move beyond mathematics, however, he begins to show some anxiety about the possibility of language finding its way back in. Before the universal algebra can be applied in a particular case, it is necessary to establish the meanings of the symbols, and it was not apparent that this could always be done without some recourse to words. Condorcet considers this a flaw in his scheme, although not a fatal one:

我们首先观察到，如果在极少数情况下，不借助一些口头解释就无法理解一个全新的理论，无法指称一个尚未被考虑过的对象，无法开展一个尚未形成任何概念的操作，那么通用语言同样不值得被冠以这个名称，同样也同样有用。这时，就会发生与口语中的情况相反的情况：有时，由于缺乏描述对象的表达式，人们不得不展示对象本身或其表象。人们需要一种语言来补充（suppléer）另一种语言。但人们可能会认为，这种缺陷在通用代数语言中非常罕见。20

We observe first that if, in a rare circumstance, it were impossible to make understood an absolutely new theory, to designate an object which had not yet been considered, to develop an operation of which one has not yet formed any idea, without having recourse to some verbal explications, the universal language would not merit less the name, would not be less useful. It would happen then, but in an opposite sense, what happens in spoken language, when sometimes one is obliged to show the object itself or its representation, because of a lack of having the expressions to describe it. One would need one language to supplement [suppléer] the other. But one might believe that this defect will not be encountered but very rarely in the language of universal algebra.20

为了使理论被理解，为了指定对象，为了形成操作的想法——这些都是在符号和一个人对世界的精神概念之间进行调解的事情。当必须调整符号和想法之间的一致性时，必须进行“言语解释”。几年后，约瑟夫·玛丽·德·热兰多 (Joseph Marie de Gérando) 认为这是一个对哲学语言的想法致命的反对意见：人们必须用现有的、大概不完美的语言来解释新造词汇的含义，从而玷污了新语言。21然而孔多塞相信在大多数情况下这不会成为问题。他认为，人们可以通过始终注意“从已知到未知”并将新想法表达为现有想法的“概括”或“限制”来大致避免语言的玷污。22这样，代数系统就可以尽可能地自给自足，而大多数单词也可以被挡在墙外。

To make theories understood, to designate objects, to form ideas of operations—these are all matters of mediating between the symbols and a person’s mental conceptions of the world. When an adjustment has to be made to the alignment between symbols and ideas, “verbal explications” must intervene. This is an objection that Joseph Marie de Gérando would judge, a few years later, to be fatal to the idea of a philosophical language: one would have to explain the meanings of the newly minted words in an existing, presumably imperfect language, thus tainting the new one.21 Yet Condorcet is confident that it will not be a problem in the majority of cases. One can mostly avoid the taint of language, he thinks, by taking care always to proceed “from known to unknown” and by expressing new ideas as “generalizations” or “restrictions” of existing ones.22 In this way, the algebraic system can be made as self-contained as possible, and words can, for the most part, be held at bay outside the walls.

孔多塞迫切地想要摆脱自然语言，这与他17世纪的前辈们对文字的不信任大致相同。但他显然担心，尽管他尽了最大努力，语言仍会重新渗透进来——他将偶尔需要法语定义视为其计划的一个“缺陷”——这在18世纪尤为明显。对于17世纪60年代的莱布尼茨和约翰·威尔金斯来说，使用文字来确立符号意义的必要性尚未成为认识论问题。两人对于如何构建符号系统持有不同观点，但在他们那个时代，人们从未怀疑过，能够独立于任何口头语言，确定地确定字符的意义。23然而，到了18世纪90年代，市场偶像的存在已变得清晰起来。不仅存在于语言中，也存在于使用这些语言的人的心中。十八世纪中叶，人们对欧洲本土语言历史的重新关注，导致了一种普遍的信念：不同的语言蕴含着不同的思维方式。这意味着，与莱布尼茨的观点相反，即使数学推理完全通过符号表达，本土语言的错误仍然会影响数学推理。要真正摆脱潜伏在脑海中的偶像，人们不仅需要新的符号，还需要新的思想。

The urgency with which Condorcet wanted to get rid of natural language manifests much the same distrust of words that had motivated his seventeenth-century precursors. But his apparent anxiety about the possibility that, despite his best efforts, language would seep back in—the fact that he sees the occasional need for French definitions as a “defect” of his plan—is distinctive of the eighteenth century. For Leibniz and John Wilkins in the 1660s, the need to use words in establishing the meanings of symbols had not arisen as an epistemological problem. The two had different views as to how the system should be constructed, but there was never much doubt in their time that it would be possible to fix the meanings of the characters with certainty, independently of any verbal language.23 By the 1790s, however, it had become clear that the idols of the market existed not just in words, but in the minds of people who spoke them. A new attention to the histories of European vernaculars in the mid-eighteenth century had led to the widespread belief that different languages embedded different ways of thinking, which implied that, contrary to Leibniz’s view, the errors of vernaculars could still affect mathematical reasoning even when that reasoning was expressed entirely through symbols. To truly dispense with the idols that lurk in one’s mind, one would need not just new symbols but also new ideas.

十八世纪数学的概念性使其风险倍增。正如阿米尔·亚历山大（Amir Alexander）所言，当时的数学家将他们的工作视为理解物质世界中数量和空间关系的一种手段。24 正如我在本章中所示，强调将方法根植于物理现实，导致了两种将符号与语言联系起来的方式的分歧。在十八世纪中叶，让-雅克·卢梭（Jean-Jacques Rousseau）赞扬了他所谓的“自然人”的美德；在卢梭看来，人性生来就有美德，但随着时间的推移，文明腐蚀了人们，文明创造了人为的欲望，使人们相互对抗。25一种解释是，这种原始主义表明需要改革。这种想法认为，我们的语言经过几个世纪的滥用已经腐化，因此我们必须重新开始，从头开始构建一种新的语言，确保避免沾染上既有的观念和态度的痕迹。正如德·热兰多所说，这种推理导致了一种“诱人的希望”，即改革语言将解决争端并消除政治不和。26孔多塞在他的通用语言方案中吸收了这种希望；这些符号被认为能够一劳永逸地解决诸如自由、平等和博爱等分裂词语的含义，从而结束席卷法国的暴力。

The stakes of this issue were heightened by the conceptual nature of eighteenth-century mathematics. As Amir Alexander has argued, mathematicians in the period viewed their work as a means of understanding quantitative and spatial relations in the material world.24 As I show in this chapter, the emphasis on grounding methods in physical reality led to a divide between two ways of relating symbols to language. In the midcentury, Jean-Jacques Rousseau praised the virtues of what he called “natural man”; for Rousseau, people were naturally virtuous but have been corrupted over time by civilization, which creates artificial wants that turn people against each other.25 On one interpretation, this primitivism suggests the need for reform. Our language has become corrupt through centuries of abuse, the thinking goes, so we must start afresh by building a new language from the ground up, being sure to avoid tainting it with traces of received conceptions and attitudes. Such reasoning led, as de Gérando put it, to the “seductive hope” that reforming language would resolve disputes and eliminate political discord.26 Condorcet picked up on this hope in his universal language scheme; the symbols were supposed to put an end to the violence that had engulfed France by settling, once and for all, the meanings of such divisive words as liberté, égalité, and fraternité.

然而，卢梭的论证也承认另一种或许堪称保守的解读。这种观点认为，最早的语言是最好的——它们最清晰、最生动、最不腐化——因此我们应该尽可能忠实于词语的原始含义。这种解读反对语言改革，认为任何语言变革都应该受到谴责，甚至可能被压制。18世纪语言思想中的这两种立场，转化为对符号方法所能实现目标的不同看法。如果说孔多塞想通过用新造的符号取代词语来使人们更加理性，那么另一种学派则将理性置于既有的语言中，并试图尽可能地保持符号与词语的日常含义相符。然而，双方都对数学的本质持有一套共同的假设。一度，数学应该基于理性，而不是任意的公理和方法，方程式必须具有含义清晰易懂。因此，符号如何定义——以及它们与词语的关系——这个问题不容忽视。

Yet Rousseau’s argument also admits another interpretation that might be called conservative. The earliest languages, the thinking goes, were the best—they were the clearest, the most vivid, the least corrupt—and so we ought to stay as true as we can to the original meanings of words. This interpretation speaks against linguistic reform, suggesting that any linguistic change ought to be lamented and, if possible, suppressed. These two positions in eighteenth-century linguistic thought translated into divergent visions for what symbolic methods could do. If Condorcet wanted to make people more rational by replacing words with newly minted symbols, another school of thought located rationality in language as it already existed and tried to keep symbols as true to the ordinary meanings of words as possible. Both sides, however, shared a set of assumptions about the nature of mathematics. For a time, mathematics was supposed to be based on reason, not on arbitrary axioms and methods, and equations had to have meanings that were clearly understood. As a result, the question of how the symbols were defined—and thus, their relation to words—could not be ignored.

本章探讨了十八世纪人们处理这个问题的各种方式。对于像孔多塞这样的激进分子来说，代数为所有交流方式提供了一个模型，一个源于理性而非腐败社会实践的语言典范。对其他人来说，符号明显独立于既定惯例是危险的。作为后一种观点的例子，我以英国数学家弗朗西斯·马塞雷斯为例，他最为人铭记的是他对负数使用的强烈反对。琼·L·理查兹等人认为，十八世纪的法国和英国数学传统对符号方法的态度不同。27但孔多塞和马塞雷斯之间的差异不仅仅是民族文化的问题；它也与语言和政治观念的冲突有关。十八世纪关于代数的争论与二十一世纪关于算法政治含义的讨论在某些方面有着惊人的相似之处，引发了诸如严格规则是否能够服务于解放目的之类的问题。然而，这些争论发生在启蒙运动的认识论中，即使计算程序已经完全机械化，它们与人类思维之间也没有明确的界限。为了说明这个问题的重要性，我先举一个数学明确地政治化的例子：土地分割。

This chapter examines the variety of ways people addressed this question in the eighteenth century. For radicals like Condorcet, algebra offered a model for how all communication should work, an example of a language derived from reason rather than from corrupt social practices. For others, the symbols’ apparent independence from established conventions was dangerous. As an example of the latter position, I consider the English mathematician Francis Maseres, who is best remembered for his vehement opposition to the use of negative numbers. Joan L. Richards and others have argued that the French and English mathematical traditions had different attitudes toward symbolic methods in the eighteenth century.27 But the difference between Condorcet and Maseres was not just a matter of national cultures; it also had to do with conflicting ideas about language and politics. The eighteenth-century disputes about algebra parallel, in some striking ways, the twenty-first-century discourse about the political implications of algorithms, raising questions such as whether strict rules can serve a liberatory purpose. Yet these disputes took place within Enlightenment epistemologies that draw no sharp line between computational procedures and human thought even when those procedures were literally mechanized. To illustrate the stakes of the issue, I begin with an instance where mathematics became expressly political: the division of land.

代数与语言的起源

Algebra and the Origin of Language

印地语-阿拉伯语的算法长期以来在测量学中得到广泛应用。“几何学”一词的词源学意义在于土地的测量，而数值计算对于根据不同形状地块的边界测量来确定其面积至关重要。测量对税收至关重要，因为国家必须确切知道一个人拥有多少土地才能确定其应纳税额。这种做法在法国大革命期间获得了新的政治共鸣。法国长期以来被划分为三十四个省，但革命开始后不久，国民制宪议会就用名为“省”（départements）的其他划分方式取代了这些省。其中一派主张建立完全正方形的省，使其形似棋盘。在该提案中，计算不再仅仅是测量已划分地块的一种方式；现在，它将在国家行政结构的重建中发挥作用。然而，该计划并未实施。在法国，美国也实施了类似的制度，其结果是，美国各郡县的形状呈现出完美的方形，这种形状至今仍存在于西方大部分地区。

The Hindi–Arabic algorism had long found a major application in surveying. The word geometry means, etymologically, the measurement of land, and numerical calculations were necessary for determining the areas of variously shaped plots based on measurements of their boundaries. Surveying was important to taxation, since the state had to know exactly how much land a person held to determine how much the person owed. This practice gained a new political resonance during the French Revolution. France had long been divided into thirty-four provinces, but shortly after the revolution began, the National Constituent Assembly replaced these provinces with other divisions called départements. One faction campaigned for the establishment of départements that were perfectly square and thus reminiscent of a checkerboard. In this proposal, calculation was no longer just a way of measuring plots of land whose divisions have already been established; now it would play a role in the reconstruction of the country’s administrative structure. Whereas this plan was not enacted in France, a version was put in force in the United States, where it resulted in the perfectly square counties that still exist in much of the West.

这种土地分配的几何模型体现了 18 世纪晚期的一种趋势，约翰·海尔布隆称之为“量化精神”。28在各种各样的背景下，数学被认为具有说服力，可以用不可否认的真理力量打破君主制和贵族权威的结构。29这样的改革面临着强大的反对。在1790年出版的《法国大革命反思》一书中，埃德蒙·伯克将棋盘方案视为革命改革压迫性质的特征。伯克写道，通过使用这种“几何分布和算术排列”，雅各宾派“把法国当作一个征服国家”。30由于棋盘计划不关注定义居民地点的传统和情感依恋，它消除了人们对自己所在地区产生个人联系的可能性。对伯克来说，这是一种暴行；但对于提倡这种方法的改革者来说，这正是关键所在——新的划分将取代阻碍平等代表权的贵族依附关系。

This geometric model of land division exemplifies a late-eighteenth-century tendency that John Heilbron calls “the quantifying spirit.”28 Across a range of contexts, mathematics was thought to have a persuasive power that could break down the structures of monarchical and aristocratic authority with the force of undeniable truth.29 Such reform faced powerful opposition. In the 1790 book Reflections on the Revolution in France, Edmund Burke takes the checkerboard scheme as characteristic of the oppressive nature of revolutionary reform. In using such a “geometrical distribution and arithmetical arrangement,” Burke writes, the Jacobins “treat France exactly like a country of conquest.”30 Since the checkerboard plan pays no attention to the traditions and emotional attachments that defined places for residents, it eliminates the possibility of feeling a personal connection to one’s region. For Burke, this is an outrage; but for the reformers who advocated such methods, it was precisely the point—the new divisions would supersede the aristocratic attachments that stood in the way of equal representation.

根据伯克等人的言论，人们可能会认为数学站在政治分歧的一边——改革者试图将数学理性强加于社会，而反革命者则试图抵制这种努力，以此来维护传统的等级制度和价值观。但现实比这更复杂。当时的政治派别不仅在如何使用数学上争论不休；他们对数学是什么也有不同的看法。在十七世纪，正如我们在第一章中看到的，托马斯·霍布斯反对代数符号构成一种“狭隘”的语言，与一个国家大多数人的说话方式格格不入。在十八世纪，符号和日常语言之间的明显脱节成为一个争论的焦点，有时预示着与既定实践的彻底决裂，有时则威胁着数学脱离现实。

On the basis of remarks like Burke’s, one might suppose that mathematics stood on one side of a political divide—that reformers were attempting to impose mathematical rationality on society, whereas counterrevolutionaries wanted to resist such efforts as a way of maintaining traditional hierarchies and values. But the reality is more complex than this. The political factions of the time were not just tussling over how to use mathematics; they also had different ideas of what mathematics was. In the seventeenth century, as we saw in chapter 1, Thomas Hobbes had objected that algebraic symbols constitute a “narrow” language that is alienated from the way most people in a country speak. In the eighteenth century, the apparent disconnect between symbols and ordinary language became a point of contention, at times promising a radical break from established practices and at others threatening to unmoor mathematics from reality.

在十八世纪，甚至在最抽象、最理论化的数学形式中，也出现了关于语言政治的立场。一个引人注目的例子出现在十八世纪中叶最伟大的数学家莱昂哈德·欧拉的著作中。欧拉的兴趣之一是计算无穷级数和的问题。一个著名的无穷级数是和，或者，我们也可以换一种说法，是。当我们把这些递减的数字相加时，和会越来越接近1，尽管它从未完全达到1；因此，这个数列被称为收敛到1，1通常被视为这个数列的和。然而，有些数列不会收敛到任何值。例如，1 − 1 + 1 − 1 + 1 − 1 + · · ·，有时也称为格兰迪级数，随着我们反复加减单位，它会永远在 1 和 0 之间交替。因此，这个级数被称为发散级数。

Positions regarding the politics of language arose in even the most abstract, theoretical forms of mathematics in the eighteenth century. A striking example appears in the work of the mid-eighteenth century’s greatest mathematician: Leonhard Euler. One of Euler’s interests was the problem of computing the sums of infinite series. A well-known infinite series is the sum , or, as we might alternatively express it, . As we add up these diminishing numbers, the total gets closer and closer to 1, although it never quite reaches it; the sequence is thus said to converge to the value of 1, which is usually seen as the sum of this series. Some series, however, do not converge on any value. For instance, the series 1 − 1 + 1 − 1 + 1 − 1 + · · ·, sometimes known as Grandi’s series, alternates between 1 and 0 forever as we repeatedly add and subtract units. This series is thus called divergent.

即使这个级数永不收敛，我们仍然会好奇，整个级数——不仅包括前n项，还包括用“+ · · ·”表示的所有无穷项——是否仍然可以得到一个和。我们可以通过将数字分成两组来合理地论证和为零：

Even though this series never converges, we might still wonder if the series as a whole—including not just the first n terms, but also the whole infinity of terms expressed by the “+ · · ·”—might still be assigned a sum. We could plausibly argue that the sum is zero by grouping the numbers in pairs:

然而，通过对它们进行不同的分组，我们可以同样合理地论证总和为 1：

By grouping them differently, however, we could argue with equal plausibility that the sum is one:

1703 年，吉多·格兰迪 (Guido Grandi) 提出该级数的真实和为 1/2，但这一结果并未得到普遍接受。31欧拉构建了他认为是格兰迪结果的明确证明，但他只能通过理论上的尝试来做到这一点。

In 1703, Guido Grandi had proposed that the true sum of this series was 1/2, but this result was not universally accepted.31 Euler constructed what he thought was a definitive proof of Grandi’s result, but he could only do so by making a theoretical gambit.

欧拉的论证发表于1760年，但很可能是在18世纪40年代发展起来的，其中涉及一种莱布尼茨先前应用于此问题的展开技巧。32这一论证通常以艾萨克·牛顿的广义二项式定理来表述，但我将以更基础的方式进行解释。我们可以从观察以下恒等式开始：

Euler’s argument, which was published in 1760 though probably developed in the 1740s, involves an expansion technique previously applied to this problem by Leibniz.32 This argument is often stated in terms of Isaac Newton’s generalized binomial theorem, but I will explain it in a more elementary way. We may begin by observing the following identity:

通过递归地将此恒等式应用于自身，我们可以将左边的表达式转换为更高阶的多项式：

By applying this identity recursively to itself, we can transform the expression on the left into polynomials of ever higher orders:

等等。从这个无限展开式中，欧拉得出结论（这一结论在现代数学中并不被接受），等价于一个无穷级数：

And so forth. From this endless expansion, Euler concludes (in a move that would not be accepted in modern mathematics) that is equivalent to an infinite series:

令a = 1；分数求值为，展开式简化为格兰迪交错级数。欧拉将此作为格兰迪论断的证明。然而，这一结果是以放弃“和”一词的通常定义作为代价的。欧拉写道，数学家传统上将无穷级数的和定义为“随着级数中项的增多，它越来越接近的量” 。33在这个定义中，发散级数显然没有和。然而，我们也可以将和定义为“生成级数的量”。34由于这个新定义（根据欧拉的说法）与收敛级数的旧定义一致，因此它没有任何缺点；并且由于它能够解决更广泛的问题，因此它更胜一筹。

Let a = 1; the fraction evaluates to and the expansion reduces to Grandi’s alternating series. Euler took this as a proof of Grandi’s claim. This result came, however, at the cost of abandoning the usual definition of the word sum. Mathematicians, Euler writes, traditionally define the sum of an infinite series as “that quantity to which it is brought closer as more terms of the series are taken.”33 In this definition, divergent series clearly do not have sums. However, we could also define the sum as “that quantity which generates the series.”34 Since this new definition coincides (according to Euler) with the old one in the case of convergent series, it has no drawback; and since it enables a wider array of problems to be solved, it is superior.

尽管欧拉的明确目标是结束关于发散级数的争论，但他的论证却在几个世纪里一直备受争议，后来甚至成为十八世纪数学被认为草率的象征。1826年，数学家尼尔斯·亨利克·阿贝尔称发散级数为“魔鬼的作品”。35就我们的目的而言，重要的不是以现代标准衡量欧拉是对是错，而是语言在他的论证中所扮演的角色。他认为，那些否认发散级数存在和的人“陷入了一场文字之争”；如果“我们改变公认的和的概念”，这些困难就会消失。36欧拉对这种修辞方法的信心，与激发孔多塞乌托邦方案的假设大致相同：人们可以通过更清晰地定义词语来解决分歧。这是否是一种可以接受的论证形式，取决于定义在多大程度上是任意的，以及这种任意性在多大程度上允许学者偏离公认的用法。

Although Euler’s explicit goal was to put an end to the disputes over divergent series, his argument remained controversial for centuries, and it later became an emblem of the perceived sloppiness of eighteenth-century mathematics. In 1826, the mathematician Niels Henrik Abel called divergent series “the Work of the Devil.”35 For our purposes, what matters is less whether Euler was right or wrong by modern standards than the role language played in his argument. Those who deny that divergent series have sums, he argues, “have stumbled into a mere battle of words”; these difficulties will vanish if “we change the accepted notion of sum.”36 Euler’s confidence in this rhetorical approach rests on much the same assumption that animated Condorcet’s utopian scheme: that one can settle disagreements by more clearly defining words. Whether this was an acceptable form of argument came down to the issue of to what extent definitions are arbitrary and to what extent that arbitrariness licensed scholars to deviate from received usages.

这些问题构成了当时语言思想界的一个主要讨论话题。孔狄亚克神父在其1746年发表的《人类知识起源论》中，对语言的发明进行了推测性描述，这种描述既是对历史的重构，也是科学界应如何重塑其语言的典范。37后世的学者们就这一描述的许多方面展开了辩论。约翰·彼得·叙斯米尔希认为，语言不可能被发明，因为如果没有某种先前存在的交流系统，就无法定义词语；因此，他得出结论，语言是上帝赐予人类的礼物。38约翰·戈特弗里德·赫尔德不同意这种观点，他对创造语言的人类智慧充满热情。39思想家们还就语言是随着时间的推移而进步还是衰落进行了辩论，卢梭认为，发达文明的语言缺乏原始文明的活力和情感。另一个问题是，究竟是平民创造了语言，还是像古怪的洛德·蒙博多认为，希腊语是由学者组成的理事会设计的。40这些起源故事包含大量没有证据的猜测，以至于语言学领域几乎否定了它们。但对于代数这样毫无争议的人造语言，起源问题却有所不同。如果语言是经过精心设计的对象，那么提出新的“和”定义就没有问题。另一方面，如果它自然而然地从平民活动中产生，这样的举动将在代数和白话语言之间建立起一道不可逾越的障碍。

Such issues formed a major topic of discussion in the linguistic thought of the time. In his 1746 Essay on the Origin of Human Knowledge, the Abbé de Condillac presents a speculative account of the invention of language that serves at once as a historical reconstruction and as a model for how sciences should remake their languages anew.37 Later writers debated a number of aspects of this account. Johann Peter Süßmilch argued that language could not have been invented, since there would be no way to define words without some preexisting system of communication; he thus concludes that language was a divine gift to humankind.38 Johann Gottfried Herder disagreed, enthusing over the human ingenuity that produced language.39 Thinkers also debated whether languages improved or declined over time, with Rousseau making a case that the languages of advanced civilizations lacked the vibrancy and emotion of primitive ones. A further question was whether commoners created language or whether, as the eccentric Lord Monboddo argued of the Greek language, it was designed by a council of the learned.40 These origin stories contained such a profusion of evidence-free speculation that the field of linguistics has all but disavowed them. But in regard to uncontroversially artificial languages such as algebra, the question of origins made a difference. If language was an object of intentional design, then there was no problem in proposing a new definition of sum. If, on the other hand, it arose naturally from the activities of commoners, such a move would create an impassible barrier between algebra and the vernacular tongue.

孔多塞侯爵的进步叙事就包含了这样一个起源故事，并阐明了其对数学符号的含义。在孔多塞看来，最初的语言是“整个社会”的创造，而非某位天才的杰作。41然而，随着文字的发明，语言后来成为社会分裂的媒介。根据孔多塞的说法，最早的书写系统是象形文字，其含义被祭司们保密，以此作为迷惑大众的手段。42从象形文字到字母文字的过渡是一种进步，因为字母表通过统一书面语言和口头语言，消除了这一社会障碍。然而，孔多塞假设，经过“一场奇怪的革命”，象形文字终有一天会回归。43孔多塞设想未来会出现另一种文字，一种“专供科学使用”的文字，可以促进“理解的精确和计算”。44在古代，这样的体系只会“延长无知”，而专门的书写形式一旦“落入哲学之手”，就能成为启蒙运动的载体。45这三个阶段的序列使得通用代数的可能性取决于社会进步。对孔多塞而言，符号只有在理性社会中产生，才是理性的。

The Marquis de Condorcet’s narrative of progress incorporates just such an origin story, and he makes its implications for mathematical symbols plain. In Condorcet’s view, the first languages were the work “of the whole society,” not of any particular genius.41 Yet language later became an agent of social division with the invention of writing. The first writing systems, according to Condorcet, were hieroglyphs, whose meanings were kept secret by the priesthood as a means of deluding the masses.42 The transition from hieroglyphs to alphabetical writing was progress because, by unifying written and spoken language, the alphabet removed this social barrier. Yet through “a strange revolution,” Condorcet hypothesizes, hieroglyphs will one day return.43 Condorcet envisions the future development of another type of writing, one “reserved exclusively for the sciences,” that could facilitate “the precise and calculated operations of the understanding.”44 Whereas in ancient times, such a scheme would only “have helped to prolong ignorance,” specialized forms of writing can become agents of Enlightenment once they lie “in the hands of philosophy.”45 This three-stage sequence makes the possibility of a universal algebra contingent on social progress. For Condorcet, symbols can only be rational if they arise in a rational society.

孔多塞坚信他的代数象形文字能够团结而非分裂人类，这源于他坚信人类的自然推理能力基本上是可靠的。用18世纪的术语来说，孔多塞是一位感性主义者：他主张一切观念都源于感官。46他的《思维概论》开篇就提出了一种强调感官数据分析的心理能力分类法：

Condorcet’s confidence that his algebraic hieroglyphs would unite rather than divide depends on a belief that the natural reasoning faculties of human beings are basically reliable. Condorcet was, in eighteenth-century terms, a sensationalist: he maintained that all ideas derive from the senses.46 His Sketch opens with a taxonomy of mental faculties that emphasizes the analysis of sensory data:

人类生来就具有接收感觉的能力；能够感知感觉，并区分构成感觉的各种简单感觉；能够记住、识别和组合感觉；能够比较这些组合；能够理解感觉的共同点和不同点；能够给感觉附加符号，以便识别感觉。更容易地组合它们，并允许随时产生新的组合。47

Man is born with the ability to receive sensations; to perceive them and to distinguish between the various simple sensations of which they are composed; to remember, recognise and combine them; to compare these combinations; to apprehend what they have in common and the ways in which they differ; to attach signs to them all in order to recognise them more easily and to allow for the ready production of new combinations.47

孔多塞后来将这种从感觉到观念再到符号的进展归功于约翰·洛克，这表明学习理解一门语言等同于学习一种理解世界的特定方式。48对感觉的分析并非完全任意；理解是“通过外部对象的作用”发展起来的。49这些对象的共享性质奠定了语言的主体间性，并使孔多塞想要实现的完美沟通变得触手可及。

The progression from sensations to ideas to signs, which Condorcet later attributes to John Locke, suggests that learning to understand a language is equivalent to learning a particular way of understanding the world.48 The analysis of sensations is not wholly arbitrary; the understanding is developed “through the action of external objects.”49 The shared nature of these objects grounds the intersubjectivity of language and places the perfect communication Condorcet wanted to achieve within reach.

这种耸人听闻的论调在语言起源的讨论中十分常见。孔狄亚克神父的著作在这场讨论中占据了重要地位。在他1749年出版的《系统论》中，他用一句格言式的、经常被引用的话总结了自己的观点：“一门运作良好的科学，仅仅是一种形式良好的语言。” 50在孔狄亚克看来，语言提供了一种“分析方法”，它的词语将世界划分为可理解的块状并表明它们之间的关系。他认为，由于科学的根本语言性质，“任何科学都应该在智慧的头脑所能理解的范围内，因为每一种形式良好的语言都是可以理解的。” 51孔狄亚克毫不掩饰地将形式良好的语言的准则用在了自己身上。“如果你不理解我，”他写道，“那是因为我不知道如何写作；如果你有时不理解我，那是因为我有时写得不好。” 52这句话与莱布尼茨的观点颇有相似之处，莱布尼茨认为，一个普遍性一旦被恰当地“整理”起来，就会迅速变得可理解。但孔狄亚克明确拒绝了莱布尼茨的先天观念学说，而是认为我们对感官对象的划分和分类方式从根本上来说是任意的。因此，语言的质量取决于其概念的切分程度。

This sensationalism is typical of the discussions of linguistic origins. The Abbé de Condillac’s work loomed large in this discourse. In his 1749 book A Treatise on Systems, he sums up his position with the gnomic, often-quoted statement that “a well-conducted science is merely a well-formed language.”50 A language, in Condillac’s view, provides a “method of analysis,” its words dividing the world into comprehensible chunks and indicating their relations. As a result of the fundamentally linguistic nature of science, he argues, “any science should be within the reach of an intelligent mind, since every well-formed language is comprehensible.”51 Condillac disarmingly turns the rubric of the well-formed language on himself. “If you do not understand me,” he writes, “it is because I do not know how to write; and if you happen sometimes not to understand me, that is because I sometimes write badly.”52 This statement bears some resemblance to Leibniz’s claim that a universal character would rapidly become comprehensible once it was properly “ordered.” But Condillac expressly rejected Leibniz’s doctrine of innate ideas, instead viewing the way we divide up and classify the objects of the senses as fundamentally arbitrary. The quality of a language, then, was measured by how sharply its concepts were cut.

至少在孔狄亚克神父后期的著作中，代数是形式良好的语言的典范。在他 1780 年去世几周后出版的《逻辑学》中，他认为用代数符号可以进行精确的证明，因为它既不受“没有确定意义的粗俗词汇”的影响，也不受“难以理解的外来词汇或野蛮词汇”的影响。53他继续说，符号代数的成功证明了“科学的进步完全取决于语言的进步”。54孔狄亚克神父和孔多塞侯爵在许多问题上持有截然不同的观点，包括代数与概念分析的等价性。据报道，孔狄亚克“憎恶”孔多塞，并指责侯爵给他端上了被污染的热巧克力，他认为这导致了他最终死于“腐烂的胆汁热”。55但很难否认他们他们都运用相同的广义思维进行思考。两者都受洛克的影响，都将感官视为知识的起源，并将代数视为一种模型，以阐明如何通过结构良好的语言来阐明我们对世界的理解。

The exemplary well-formed language, at least in the Abbé de Condillac’s later writings, is algebra. In Logic, published a few weeks after his death in 1780, he argues that exact proofs are possible in algebraic notation because it is tainted by neither “vulgar words that have no determinate sense” nor “foreign or barbarous words that are poorly understood.”53 The success of symbolic algebra, he continues, proves that “scientific progress depends solely on the progress of languages.”54 The Abbé de Condillac and the Marquis de Condorcet had disparate views on a number of matters, including the equivalence of algebra to conceptual analysis. Condillac reportedly “detested” Condorcet and blamed the marquis for serving him the tainted hot chocolate that, he believed, caused the “putrid bilious fever” from which he eventually died.55 But it is hard to deny that they were thinking within the same broad terms. Both were under the sway of Locke, both took the senses as the origin of knowledge, and both regarded algebra as a model for how a well-constructed language can clarify our understanding of the world.

这种思维方式提供了一种理解符号系统功能的方式，这种功能在十八世纪独具特色。在这一时期，代数学家将他们的科学视为对物质属性的高度抽象的表征。正如丹尼斯·狄德罗所写：“所有抽象概念都仅仅是缺乏具体意义的符号。所有抽象科学都只是在玩弄符号。当符号脱离物理对象时，精确的图像就消失了；只有当符号与物理对象重新结合在一起时，科学才能再次成为真实事物的问题。” 56这种坚持将数学建立在感性世界之上的理念，立刻点燃了人们对一种以代数为模型的通用语言的希望，也引发了一些理论问题。是什么赋予了代数如此卓越的清晰度？它真的如此清晰吗？这些问题在十八世纪引起了争议，而答案具有政治意义——它们影响着孔多塞试图进行的那种合理化改革的合法性。

Such thinking provided a way of understanding the function of symbolism that was distinctive of the eighteenth century. In this period, algebraists viewed their science as a highly abstract representation of the properties of material things. As Denis Diderot wrote, “Every abstraction is merely a symbol devoid of particularized meaning. Every abstract science is simply juggling with symbols. The exact picture was dropped when the symbol was separated from the physical object, and it is only when the symbol and the physical object are brought together again that the science once again becomes a matter of real things.”56 This insistence on grounding mathematics in the sensible world at once stoked the hope of a universal language modeled on algebra and raised theoretical questions. What was it that gave algebra its superior clarity? And was it really so clear after all? These issues were up for dispute in the eighteenth century, and the answers had political stakes—they affected the legitimacy of the sort of rationalizing reform that Condorcet was trying to undertake.

一种特殊的语言

A Particular Kind of Language

尽管代数在十八世纪体现了清晰的表达方式，但其倡导者却一直被一种挥之不去的怀疑所困扰，认为它实际上毫无意义。在《哲学要素》（1759 年）一书中，让·勒朗·达朗贝尔 (Jean Le Rond d'Alembert) 提出了一个问题：为什么尽管代数的原理和推论确定无疑，但“在某些方面仍未完全摆脱晦涩难懂的困境”。57他以负数为例，并写道，他不知道有哪部著作能提供关于负数的清晰理论。58达朗贝尔指的是符号表示法的广泛应用所导致的最臭名昭著的概念调整之一。著名的波特-罗亚尔逻辑的作者之一安托万·阿诺 (Antoine Arnauld) 在 1667 年的一本几何教材中指出，从直觉上讲，较大数与较小数的比例应该大于反之。59然而，他指出，负数的情况并非如此，至少在符号代数中是这样：。这远非关于负数的唯一概念难题；欧拉在其关于发散级数的文章中，考虑了这样的建议：的值应该高于无穷大，因为它涉及除以小于零的数。60

As much as algebra exemplified perspicuous expression in the eighteenth century, its advocates were haunted by a lingering suspicion that it did not actually make sense. In Elements of Philosophy (1759), Jean Le Rond d’Alembert asks why algebra, in spite of the certainty of its principles and inferences, “is not yet entirely exempt from obscurity in certain regards.”57 As a specific example, he offers negative numbers, of which, he writes, he does not know a single work that provides a clear theory.58 D’Alembert is referring to one of the most notorious conceptual readjustments caused by the widespread adoption of symbolic notation. Antoine Arnauld, one of the authors of the well-known Port-Royal Logic, argued in a 1667 geometry text that, intuitively, the proportion of a larger number to a smaller one should be larger than the reverse.59 Yet, he points out, this is not the case with negative numbers, at least in symbolic algebra: . This was far from the only conceptual conundrum regarding negatives; in his article on divergent series, Euler considers the suggestion that the value of should be higher than infinity because it involves dividing by something less than zero.60

如何解释这些悖论，一直是法国哲学家们争论不休的问题。采用符号方法从根本上改变了数量的定义，导致符号与古典数论之间出现认识论鸿沟。达朗贝尔认为，代数是“一种语言，它和其他语言一样，具有形而上学”。61 正如盲目遵循语法规则并不能理解语言一样，“庸俗的人只庆祝代数方法的结果”，而真正开明的人也能“看到产生结果的萌芽”。62要理解代数证明的含义，必须学会用代数的方式思考，否则，符号只会导致混乱。这一论点更普遍地狭窄了代数在知识中的作用。对达朗贝尔而言，代数分析必须特别区别于逻辑分析——即将概念划分为组成部分——因为混淆两者会掩盖代数在数量概念中的基础。

What to make of such paradoxes was a matter of dispute among the French philosophes. The adoption of symbolic methods fundamentally altered the definition of quantity, leading to an epistemological gap between the symbols and classical theories of number. As an explanation of this gap, d’Alembert concludes that algebra is “a kind of language which has, like the others, its metaphysics.”61 Just as blindly following the rules of grammar does not lead to an understanding of language, “the vulgar only celebrate the result” of algebraic methods, whereas the truly enlightened can also “see the germ that produced it.”62 To perceive the meanings of algebraic proofs, one must learn to think algebraically; otherwise, the symbols will only lead to confusion. This argument assigned algebra a narrow role within knowledge more generally. For d’Alembert, algebraic analysis must especially be distinguished from logical analysis—that is, the division of concepts into their component parts—since conflating the two would obscure algebra’s basis in notions of quantity.

孔狄亚克神父坚决反对这种想法。在《逻辑学》中，他指出：“我不同意那些声称代数是一种语言的数学家。我认为它是一种语言，不可能是别的任何东西。” 63正如 WR Albury 指出的那样，他所指的两位数学家是达朗贝尔和阿莱克西斯-克洛德·克莱罗，后者在其 1746 年出版的《代数要素》一书中将代数符号称为一种“特殊类型的语言”，它使用简单的符号，使运算更容易一目了然。64代数是“一种语言”和“一种特殊类型的语言”之间的区别可能看起来很迂腐，但这其中涉及到一些重要的事情。针对有人反对孔狄亚克对“推理”的描述仅仅捕捉了“我们在数学中推理的方式，即推理是通过方程式进行的”，孔狄亚克断言“方程式、命题和判断本质上是同一件事物，因此，我们在所有科学中都以同样的方式进行推理”。65因此，词语和代数符号之间的区别是表面的：“我们不应该认为科学是精确的，或者我们只能在使用x、a和b时才能严格证明”。66这一论点标志着符号与语言关系的逆转。17 世纪，塞斯·沃德强烈坚持认为符号不是语言；一个世纪后，孔狄亚克同样强烈地坚持认为符号是一种语言。这一坚持的关键在于符号是否真的能够清晰地传达思想，或者相反，它们是否会混淆事物。

The Abbé de Condillac took an emphatic stand against this thinking. In Logic, he states, “I do not agree with mathematicians who claim that algebra is a kind of language. I say it is a language and cannot be anything else.”63 As W. R. Albury points out, two of the mathematicians he means are d’Alembert and Alexis-Claude Clairaut, who, in his 1746 book Elements of Algebra, calls algebraic notation a “particular kind of language” that uses simple signs to make it easier to see one’s operations at a glance.64 The distinction between algebra being “a language” and being “a particular kind of language” might seem pedantic, but something serious is at stake. Against the objection that his description of “reasoning” only captures “the way we reason in mathematics, where reasoning is carried out with equations,” Condillac asserts that “equations, propositions, and judgments are at bottom the same thing, and that consequently we reason the same way in all the sciences.”65 As a result, the difference between words and algebraic symbols is superficial: “We should not suppose that the sciences are exact—or that we prove rigorously—only when we use x’s, a’s, and b’s.”66 This argument marks a reversal in the relation of symbols to language. In the seventeenth century, Seth Ward was vehemently insisting that symbols were not a language; a century later, Condillac was insisting with equal vehemence that they were a language. At stake in this insistence was whether the symbols really did have the ability to convey ideas clearly or whether, to the contrary, they confused things.

实际上，当时的代数并不总是像孔狄亚克神父所断言的那样与语言完美对应。从18世纪70年代初开始，意大利裔法国数学家约瑟夫-路易·拉格朗日就一直试图系统化和推广其前辈们的方程求解方法。67 1795年，拉格朗日在巴黎师范学院举办了一系列讲座，试图澄清自从卡尔达诺在1540年代遇到三次方程的难题以来，三次方程就一直存在着诸多难题。拉格朗日认为，即使这些表达式“不易进行数值计算”，它们仍然可以“用于代数运算”，作为解决问题的一种手段。68他在其他地方指出：“指责代数学无法得出具有该问题所有普遍性的结果，这对代数学来说是极其不公的。唯一的前提是能够读懂代数学独特的笔迹，这样我们就能从中看到它本质上所能包含的一切。” 69他特别指出，数学家们忽略了立方根有三个值的事实，一些是实数，一些不是。拉格朗日接受这些值的理由取决于代数量（它们通过诸如√之类的运算相互关联来定义）与普通的数字概念之间的区别。70正如达朗贝尔所说，代数有其自己的形而上学，不同于人们在其他科学中的思维方式。

In practice, the algebra of the period did not always correspond as neatly to language as the Abbé de Condillac asserted. Starting in the early 1770s, the Italian French mathematician Joseph-Louis Lagrange had been attempting to systematize and generalize the equation-solving methods of his precursors.67 In 1795, Lagrange delivered a series of lectures at the École Normale in Paris, in the course of which he attempted to clear up the difficulties that had surrounded cubic equations since Cardano came across the troublesome in the 1540s. Even though such expressions are not “susceptible of being numerically evaluated,” Lagrange suggests, they can still “be employed in the operations of algebra” as a means of solving problems.68 “It would,” he states elsewhere, “be the height of injustice to algebra to accuse it of not yielding results which were possessed of all the generality of which the question was susceptible. The sole requisite is to be able to read the peculiar hand-writing of algebra, and we shall then be able to see in it everything which by its nature it can be made to contain.”69 He notes, in particular, that mathematicians have overlooked the fact that cube roots have three values, some real and some not. Lagrange’s rationale for accepting such values depends on a distinction between algebraic quantities, which are defined by how they relate to each other through operations such as √, and ordinary conceptions of number.70 Algebra, as d’Alembert might say, had a metaphysics of its own, different from how people think in other sciences.

尽管存在诸多困难，十八世纪晚期的量词学家大多认同孔狄亚克神父对数学论证普遍可理解性的乐观态度。尽管孔多塞侯爵在分析的本质问题上倾向于达朗贝尔的观点，但他在其最具影响力的“社会数学”著作之一《论分析在多数决策概率中的应用》（1785年）中，巧妙地将数学等同于理性。在这本书中，他发展了一种量化决策机构（例如陪审团）可靠性的方法，该方法基于陪审团的规模及其成员的“启蒙”程度。孔多塞意识到书中繁复的符号会给读者带来困难，因此他在一篇占全书一半以上篇幅的“开场白”中，以通俗易懂的方式解释了他的论点。这样一来，他写道：“即使不是几何学家的读者，在评判该著作时，也只需承认所给出的计算证明为真即可。” 71尽管不懂数学的读者也必须相信孔多塞的证明是正确的，但孔多塞对结果并不抱太大期望：“几乎在任何地方，人们都会发现结果与最简单的推理一致；但是诡辩和空洞的诡辩很容易掩盖推理，因此，如果我所做的只是用数学证明的权威支持一个有用的真理，我就感到高兴了。” 72其假设是，即使人们无法理解计算的复杂性，他们的结论仍然直观地正确；证明的目的不在于产生新的结果，而在于防范导致人们犯错的腐败力量。

In spite of such difficulties, the quantifiers of late eighteenth century mostly shared the Abbé de Condillac’s optimism about the universal comprehensibility of mathematical arguments. Although he tended to side with d’Alembert on the nature of analysis, the Marquis de Condorcet rhetorically equates mathematics with reason in one of his most influential works of “social mathematics,” Essay on the Application of Analysis to the Probability of Majority Decisions (1785). In this book, he develops a means of quantifying the reliability of decision-making bodies such as juries based on their size and the degree of “enlightenment” of their members. Recognizing the difficulty that the book’s thickets of symbols will present for readers, Condorcet also explains his argument nontechnically in a “Preliminary Discourse” more than half the length of the main text of the book. This way, he writes, “readers who are not Geometers, will only need, in order to judge the work, to admit as true that which is given to be proven by calculation.”71 Even though the innumerate reader will have to take Condorcet’s word that the proofs are correct, Condorcet does not expect much surprise as to the results: “Almost everywhere one will find results consistent with what the simplest reason would have dictated; but it is so easy to obscure reason by sophisms and by vain subtleties, that I would consider myself happy when I have done nothing but support by the authority of a mathematical demonstration one single useful truth.”72 The assumption is that, even if one cannot follow the intricacies of the computations, their conclusions will remain intuitively true; the purpose of the proofs is not to produce new results but rather to guard against the corrupt forces that lead people into error.

类似的想法也可以在孔多塞侯爵的通用语言项目中看到。正如贝克所指出的，孔多塞在这个项目中的目标是将数学分析的确定性扩展到通常属于其他不太可靠的分析形式的研究领域。73贝克指出，这样做使他想到了机械化逻辑推理的可能性。74然而，孔多塞的思想与现代算法思维的工具化机械化方法之间存在重要区别。侯爵在晚期残篇中写道，机械推理工具“会显得荒谬”，直到它们所基于的分类“表”被经验证明是有价值的。75他的通用代数建立在这样一种信念之上：这些表格可以足够有条理地开发，以至于人们可以轻松地就类别的含义达成一致。如果这个前提被证明是错误的，那么这个系统就根本不符合他的平等政治；相反，他的数学象形文字就像埃及祭司的秘密语言一样，对大多数人来说都是一个难以理解的谜，只有少数人知道。

A similar thinking may be discerned in the Marquis de Condorcet’s universal language project. As Baker points out, Condorcet’s goal in this project was to extend the certainty of mathematical analysis to areas of inquiry that were ordinarily the domain of other, less reliable forms of analysis.73 Doing so led him, as Baker points out, to the possibility of mechanizing logical inference.74 Yet there is an important difference between Condorcet’s thinking and the instrumental approach to mechanization that characterizes modern algorithmic thinking. Mechanical implements for reasoning, the marquis writes in a late fragment, “would appear ridiculous” until the “tables” of classification on which they are based are proven worthy by experience.75 His universal algebra fed on a faith that these tables could be developed methodically enough that there would be no problem bringing people to an accord about what the categories meant. If this premise turned out to be false, then the system would not fit with his leveling politics at all; instead, his mathematical hieroglyphs would, like the secret language of the Egyptian priesthood, become an impenetrable mystery to all but a few.

沿着这一思路的最有力的论据出现在德意志各邦。在孔多塞写下手稿的几十年前，语言学家约翰·戴维·米夏埃利斯在他的法文版《论观点对语言的影响以及语言对观点的影响》中详细且论证充分地反对哲学语言体系。这篇论文于 1759 年获得了普鲁士学院奖，并在国际上广受好评，获得了达朗贝尔的高度赞扬。76该论文的法文译本用大量篇幅反对“博学语言”的概念，显然是在暗指莱布尼茨的普遍性特征。77他在这个后来对德国及其他地区产生重大影响的论证中得出结论：尽管语言带来了种种问题，但我们在很大程度上还是被它束缚着——莱布尼茨试图废除文字，但其代价大于收益。

The strongest arguments along these lines appeared in the German states. A few decades before Condorcet wrote his manuscript, the philologist Johann David Michaelis had made a detailed, well-argued case against philosophical language schemes in the French edition of his Dissertation on the Influence of Opinions on Language and of Language on Opinions. This essay won a Prussian Academy prize in 1759, and it was well received internationally, garnering high praise from d’Alembert.76 The French translation of the essay includes an extensive section arguing against the idea of a “learned language,” with apparent reference to Leibniz’s universal characteristic.77 In an argument that would go on to have significant influence in the German context and elsewhere, he concludes that, in spite of all the problems language causes, we are to a large extent stuck with it—the costs of doing away with words, as Leibniz had attempted to do, outweigh the benefit.

米凯利斯的一个论点也涉及到孔多塞提出的一个问题：一门学术语言的潜在分裂性。米凯利斯认为，一种普遍性特征会在博学之人与未受过教育之人之间筑起一道屏障；它可以被用来掩盖新发明的运作方式，从而将其描绘成“虚假的奇迹”，以此来欺骗大众。78在学术写作中，“作者将他所使用的专业语言视为至高无上的主人。他说，这就是我赋予这个术语的含义，这就是我赋予它的定义：那么，我们都有义务去理解他，正如他宣称他将被理解一样，我们几乎无法与他争论这一权利，就像规定代数学家应该将什么线条称为a和什么b。” 79与这种“神圣的暴政”相反，在白话文中“一切都是民主的：词语不能被剥夺其既定的意义，除非经过人民的同意，并通过一点一点引入的相反用法。” 80一门学来的语言对于所有国家来说都是陌生的，它永远不可能像我们从摇篮里听到的语言那样被深入吸收。

One of Michaelis’s arguments regards an issue Condorcet raised as well: the potentially divisive quality of a learned language. A universal characteristic, Michaelis argues, would throw up a barrier between the learned and the uneducated; it could be used to delude the masses by concealing the way new inventions work and thus casting them as “false miracles.”78 In learned writing, “an author treats as sovereign master the technical language he makes use of. He says, this is the meaning I fix to this term, this is the definition I give of it: we then are all obliged to understand him, as he has declared he will be understood, and as little can we contest that right with him, as prescribe to the Algebraist what lines he shall call a and what b.”79 In contrast to this “sacred tyranny,” in vernaculars “all is Democratic: words cannot be deprived of their received meaning but by the consent of the people, and by a contrary usage, that is introduced bit by bit.”80 Being foreign, as it were, to all nations, a learned language could never be absorbed as deeply as we absorb the languages we hear from the cradle.

米凯利斯的文章提出了一种对语言缺陷的不同回应，这种回应在本世纪末产生了巨大的影响：富有想象力的文学既能诉诸情感，又能诉诸理性，比莱布尼茨的特色或孔多塞的代数等明确规定性的项目更适合影响一个民族的语言实践。81随着德国古典主义的兴起，这种思路在本世纪末在德国占据主导地位。这场运动的关键人物是JG赫尔德，他区分了“学者的文化和人民的文化”：

Michaelis’s essay suggests a different response to the flaws of language that would gain a great deal of influence by the end of the century: that imaginative literature, in appealing to the feelings as well as to reason, was better suited to influence the linguistic practices of a people than explicitly prescriptive projects such as Leibniz’s characteristic or Condorcet’s algebra.81 This line of thought would gain dominance in Germany around the end of the century with the emergence of German classicism. A key figure in this movement was J. G. Herder, who draws a distinction between “the culture of the men of learning and the culture of the people”:

因此，代数是一门神秘的科学，因为在欧洲，很少有人理解它，尽管学习它并不被禁止。如今，我们确实以一种无用且有害的方式，在许多方面混淆了学术文化和大众文化的领域，从而将后者的范围几乎扩大到前者；古代国家的奠基者以更人性化的视角思考，在这个问题上也思考得更明智。他们将人民的文化根植于健全的道德和实用的艺术；他们认为，人民既没有资格，也不可能从宏大的理论中受益，即使是哲学或宗教理论也是如此。82

So it is that algebra is an occult science, for few in Europe understand it, though learning it is prohibited to none. Now we have indeed, in a useless and damaging manner, in many respects confused the spheres of learned and popular culture, thereby extending the range of the latter almost to that of the former; the ancient founders of states, who thought in more human terms, thought more wisely on this subject also. They rooted the culture of the people in sound morals and useful arts; they deemed the people neither qualified nor likely to benefit by grand theories, even in philosophy or religion.82

与米凯利斯一样，赫尔德将代数与自然语言区分开来。自然语言在群体中有机地发展，并且能够抵抗任何刻意改变的行为。在赫尔德看来，如果像孔多塞那样，认为一个人能够完全将母语的影响从自己的思维方式中抹去，就等于忽视了群体成员身份在人类意识发展中所起的关键作用。

Like Michaelis, Herder distinguishes algebra from natural languages, which develop organically in communities and have the ability to resist intentional efforts to alter them. In the Herderian view, to suppose that one could entirely expunge the influence of one’s mother tongue from one’s way of thinking, as Condorcet wished to do, was to overlook the critical role that membership in a group plays in the development of human consciousness.

米凯利斯和赫尔德的论证指出了孔多塞等方案的一个实际问题：它们不仅需要设计一套符号系统，还需要训练人们以新的方式思考。如何解决这个问题取决于人们期望科学服务于何种社会功能。一种选择是简单地接受学术领域与大众领域之间的分离；这本质上是莱布尼茨的观点，他不介意将科学知识限制在特定人群中。然而，孔多塞对这种划分并不满意。他写道，阻碍进步的最古老障碍之一是“将人类分裂为两部分：一部分人注定要教导，另一部分人则要信仰”。83在他看来，象形文字与口语的疏离加剧了这种分裂。孔多塞的代数象形文字并非要让我们回到这种古老的境地。其目标是启迪整个社会。要实现这一目标，不仅需要发展语言本身，还需要支持它的基础设施——可以教人们阅读符号的教育机构和可以广泛传播知识的自由媒体。实现通用代数意味着将科学思想灌输到每个人的头脑中。

The arguments of Michaelis and Herder pointed out a practical problem with schemes like that of Condorcet: they would require not just designing a system of symbols but also training people to think in a new way. How one addressed this problem depended on what social function one expected science to serve. One option was simply to accept the disconnect between the learned and popular spheres; this was essentially the view of Leibniz, who had no problem restricting knowledge of science to a select group of people. Condorcet, however, was not comfortable with such a division. One of the most ancient hindrances to progress, he writes, is “the separation of the human race into two parts: the one destined to teach, the other for believing.”83 This separation, according to him, was widened by the alienation of hieroglyphs from spoken language. Condorcet’s algebraic hieroglyphs were not supposed to return us to this ancient situation. The goal was to enlighten the whole society. Achieving this end would require not only the development of the language itself but also an infrastructure to support it—educational institutions that could teach people to read the symbols and a free press that could disseminate knowledge widely. Bringing the universal algebra into being would mean installing scientific ideas in every mind.

然而，这一革命性纲领并非莱布尼茨精英主义的唯一替代方案。如果孔多塞想用更科学的东西取代语言，也可以反其道而行之，努力使学术论述植根于民众的语言之中。84赫尔德（至少有时）持这种立场，他写道，通过推行弗朗西斯·培根提出的从语言中清除“偶像”的方案，“一个人剥夺了父辈们数个世纪以来的援助，赤身裸体地站在那里” 。85孔多塞将白话文视为错误和偏见的宝库，而赫尔德则将其视为我们从祖先那里继承下来的“宝藏”。86类似的想法促使伯克捍卫传统的土地分割制度。从这个角度来看，孔多塞试图通过用他视为“开明”的做法取代他所认为的“野蛮”做法，来实施某种文化帝国主义，并试图以此将一套特定的价值观强加于世界。这些论点或许可以与以赛亚·伯林所称的反启蒙运动联系起来，反启蒙运动重视文化传统，并对孔多塞试图实施的合理化方案表示怀疑。87

This revolutionary program was not, however, the only alternative to Leibniz’s elitism. If Condorcet wanted to replace language with something more scientific, one could also go the other way, striving to keep learned discourses grounded in the language of the people.84 This was (at least at times) the position of Herder, who wrote that, by pursuing Francis Bacon’s program of expunging “idols” from language, “one has stripped oneself of the aid of all the centuries of one’s fathers, and stands there naked.”85 Whereas Condorcet viewed the vernacular as a repository of errors and prejudices, Herder regarded it as a “treasure” that we inherit from our ancestors.86 Similar thinking motivated Burke’s defense of traditional land divisions. Condorcet was, from this standpoint, attempting to enact a sort of cultural imperialism by replacing practices he judged “barbaric” with those he deemed “enlightened” and, by doing so, trying to force a particular set of values on the world. These arguments might be identified with what Isaiah Berlin called the Counter-Enlightenment, a movement that valued cultural traditions and looked askance at the program of rationalization that Condorcet was trying to enact.87

然而，我们不应过于简单地认为“启蒙运动”和“数学”在十八世纪总是一致的。即使是赫尔德，尽管他更喜欢情感而非冷酷的理性，也基于约翰·海因里希·兰伯特的数学著作，提出了一系列用于历史和伦理的伪代数“定律”。88这一时期也有一些数学家重视传统和语言的连续性，他们对符号的使用也反映了这些价值观。一个臭名昭著的例子是弗朗西斯·马塞雷斯。作为一名狂热的辉格党人，马塞雷斯支持一些自由主义事业，例如废除奴隶制，但在法国大革命后，他坚决站在伯克一边；1790年，他抱怨在法国扎根的“高度民主精神”。89他的数学著作的特点是，他英勇地努力使代数与……保持一致他认为这是英国绅士的常识，以此来抵抗孔多塞等激进分子试图通过符号手段打破日常语言的束缚。

One should not, however, assume too simplistically that “the Enlightenment” and “mathematics” were always aligned in the eighteenth century. Even Herder, for all that he preferred feeling over cold rationality, propounded a series of pseudoalgebraic “laws” for history and ethics based on the mathematical work of Johann Heinrich Lambert.88 There were also mathematicians in the period who valued tradition and linguistic continuity and whose use of symbols reflected these values. A notorious example is Francis Maseres. An ardent Whig, Maseres supported some liberal causes, such as the abolition of slavery, but he sided strongly with Burke in the wake of the French Revolution; in 1790, he complained of the “highly democratical spirit” that had taken root in France.89 His mathematical work is characterized by a heroic effort to keep algebra aligned with what he took to be the common sense of English gentlemen. By doing so, he put up resistance against the break from ordinary language that Condorcet and other radicals were intentionally trying to create by means of symbols.

纯粹的废话和难以理解的术语

Mere Nonsense and Unintelligible Jargon

马塞雷斯如今在思想史上已是边缘人物，但他比人们普遍认为的更有时代代表性。马塞雷斯在学术上并不成功，未能竞选剑桥大学卢卡斯数学教席，但他出版了几本数学著作，获得了各方面的赞誉。90从1766年到 1769 年，他担任魁北克总检察长，从 1773 年起，他担任财政大臣游击男爵，该职位负责征收皇家税收。马塞雷斯属于一个数学思想流派，该流派还包括罗伯特·西姆森和威廉·弗伦德，该流派认为负数的概念问题是它们毫无意义的证据。91这个思想流派向那些反对启蒙运动重塑语言计划的人展示了一种以概念为基础的代数学是什么样子的。

Maseres is now a marginal figure in intellectual history, but he was more typical of his time than is generally recognized. Maseres was not an academic success, losing a bid for the Lucasian Chair of Mathematics at Cambridge, but he published several mathematical books that garnered praise from various quarters.90 From 1766 to 1769, he served as the attorney general of Quebec, and from 1773 on he held the post of cursitor baron of the Exchequer, a position involved in the collection of royal revenues. Maseres was part of a school of mathematical thought, also including Robert Simson and William Frend, that viewed the conceptual problems surrounding negative numbers as proof that they were simply meaningless.91 This school of thought illustrates what an algebra grounded in concepts looked like for those who opposed the Enlightenment project of remaking language.

马塞雷斯的第一部主要数学著作是他 1758 年的《论代数中负号的使用》。他写道，这本书“试图以几何书籍通常认为必要的那种清晰和准确的推理来对待代数学，但由于原因不太容易确定，这在代数学书籍中几乎普遍被忽视。” 92马塞雷斯和孔多塞虽然不同，但他们都对符号方法的潜在分裂性感到警惕，并且相信改善的最大希望在于感官。正如马塞雷斯所说，代数学家很容易“自以为有意义，而实际上他没有意义”，因为符号论证只针对理解，而不是（像几何学一样）同时针对理解和感官。93与孔多塞一样，马塞雷斯也表达了对难以捉摸的符号可能被用来欺骗信息匮乏者的担忧。马塞雷斯指出，用文字或符号比用图表更容易撒谎；“在不涉及感官的情况下，人们更容易被欺骗。” 94

Maseres’s first major mathematical work was his 1758 Dissertation on the Use of the Negative Sign in Algebra. This book, he writes, is “an attempt to treat the Science of Algebra with the same perspicuity and accuracy of reasoning that has usually been thought necessary in books of Geometry, but which, through causes not very easy to be fixed upon with certainty, has been almost universally neglected in books of Algebra.”92 Different as Maseres and Condorcet were, they shared a wariness of the potentially divisive nature of symbolic methods as well as a belief that the best hope for amelioration lies in the senses. An algebraist, as Maseres has it, may easily “fancy he has a meaning where, in reality, he has none” because symbolic arguments address themselves only to the understanding, not (as in geometry) to both the understanding and the senses.93 Like Condorcet, Maseres expresses the concern that inscrutable symbols could be used to delude the underinformed. It is, Maseres states, easier to lie with words or symbols than with diagrams; “wherein the senses are not concerned, men are much more easily deceived.”94

对马塞雷斯来说，将代数建立在意义之上意味着消除负数的概念。他在书的开篇就宣称：“关于负量，最清晰的概念……是从一个大于自身的量中减去一个量。” 95根据这个定义，孤立地考虑的量没有符号，因此，像 −5 × −5 = +25 这样的负数计算规则“纯粹是无稽之谈，难以理解的术语”。96对此类规则的不满并不罕见。就当时而言。正如我们所见，当时关于用负数进行数学运算时应该如何进行存在着广泛的争论；用负数除似乎会引发概念上的难题，而负数对数的规则也一直悬而未决。马塞雷斯的理论旨在通过从一开始就防止问题的出现来解决所有这些困惑。没有负数；只有减法，所有代数都应该遵循易于理解的正数概念。

Grounding algebra in the senses means, for Maseres, eliminating the idea of a negative number. He begins the book with the declaration that “the clearest idea that can . . . be formed of a negative quantity is that of a quantity that is subtracted from another greater than itself.”95 Under this definition, a quantity considered in isolation has no sign, and so rules for computation with negatives, as in −5 × −5 = +25, are “mere nonsense and unintelligible jargon.”96 A dissatisfaction with such rules was not unusual for the time. As we have seen, there were extensive debates in the period about what should happen when one performs mathematical operations with negative quantities; dividing by a negative seemed to raise conceptual conundrums, and the rules for logarithms of negatives remained unsettled. Maseres’s theory aims to resolve all this confusion by preventing the issue from arising in the first place. There are no negative quantities; there is merely subtraction, and all of algebra is supposed to follow from the easily intelligible idea of a positive number.

这种概念方法体现了对数学与其他形式知识相结合的重视，琼·L·理查兹认为这是英国数学的一个特点。97马塞雷斯尤其力求保持符号方法和词语通常含义之间的连续性。他写道，乘法的主要含义“显然是重复，或者将一个量重复一定次数”；后来，这个词也适用于分数，但他声称，只是带有某种程度的扭曲。98这种伪语言学思维与达朗贝尔在其百科全书条目开头经常使用的词源学思考并无太大区别。事实上，马塞雷斯对负数的解释与达朗贝尔的解释也并非完全不同：他们都认为负数只有相对于其他量才有意义。但与这位法国数学家不同，马塞雷斯并不试图构建一种比“庸俗”的数量概念更精确的新“形而上学”。相反，他正着手根据（他认为）其国家日常语言中已经存在的真实概念来证明代数方法的有效性。

This conceptual approach exemplifies an emphasis on integrating mathematics with other forms of knowledge that Joan L. Richards has identified as a characteristic of English mathematics.97 Maseres sought, in particular, to maintain continuity between symbolic methods and the ordinary meanings of words. The primary sense of multiplication, he writes, “is evidently repetition, or the taking a quantity over and over again a certain number of times”; the word later came to be applied to fractions, but only, he claims, with some degree of distortion.98 This pseudophilological thinking is not especially different from the etymological musings with which d’Alembert often begins his encyclopedia articles. Nor, indeed, is Maseres’s explanation of negatives wholly distinct from d’Alembert’s: both thought that negatives only made sense relative to some other quantity. But unlike the French mathematician, Maseres is not seeking to construct a new “metaphysics” that is more exact than “vulgar” notions of quantity. Instead, he is setting out to prove the validity of algebraic methods based on the true notions that already existed (he thinks) in the everyday language of his country.

从现代视角来看，马塞雷斯的书似乎并没有阐明问题。他坚持避免使用负数，导致不同情况的规则激增，使得代数比（从我们的角度来看）它应有的复杂程度要高得多。例如，如果a > b，则必须采用不同的程序来计算c + a − b和c − a + b。在第一种情况下，可以按顺序执行运算；但在后一种情况下，必须颠倒顺序，先将b加到c上，然后再减去a。99当马塞雷斯转向解方程时，事情变得更加复杂。正如罗伯特·雷科德和杰罗拉莫·卡尔达诺在 16 世纪的著作中所述，马塞雷斯将二次方程分为多种类型，例如xx + px = r、xx − px = r和px − xx = r。100由于系数不能为负，这些方程不能互相转化；因此马塞雷斯必须为每个方程给出不同的解法规则。

From a modern perspective, Maseres’s book hardly seems to clarify things. His insistence on avoiding negatives results in a proliferation of different rules for different cases, making algebra far more complex than it needs (from our perspective) to be. For instance, if a > b, one must employ a different procedure for computing c + a − b and c − a + b. In the first case, one may perform the operations in order; but in the latter case, one must reverse the order and add b to c before subtracting a.99 Things become even more complex when Maseres moves on to solving equations. As in the sixteenth-century texts of Robert Recorde and Gerolamo Cardano, Maseres divides quadratic equations into multiple types, such as xx + px = r, xx − px = r, and px − xx = r.100 Since coefficients cannot be negative, these equations cannot be converted into one another; hence Maseres has to give different rules for how to solve each one.

由于这种看似不必要的复杂化，马塞雷斯经常被描绘成代数史上的反派，一个阻碍进步的反动分子。101他所要表达的意思在一个相对被忽视的这是他工作的一部分。1783 年，他出版了一本书，可以与迈克尔·达里关于复利的小册子一起归入计算手册的类别。这本名为《人寿年金学说原理》的书介绍了一系列计算人寿年金价值的方法，人寿年金是一种从给定的起始日期到人寿生命结束分期支付的投资。马塞雷斯出版这本书是为了试图恢复一项失败的议会法案，该法案将允许穷人从教堂管理员那里购买年金，从而减少因年龄过大而无法工作的穷人所需的国家援助数额。102马塞雷斯的程序平淡无奇，但却无疑是算法性的：

On account of this seemingly needless complication, Maseres is often cast as a villain in histories of algebra, as a reactionary holding back progress.101 What he was getting at becomes clearer in a relatively neglected part of his work. In 1783, he published a book that one might place alongside Michael Dary’s pamphlet on compound interest in the genre of computational manuals. This book, titled The Principles of the Doctrine of Life-Annuities, presents a series of methods for computing the values of life annuities, which are investments that pay out in installments from a given start date to the end of a person’s life. Maseres published it as part of an attempt to revive a failed parliamentary bill that would have enabled the poor to purchase annuities from church wardens, thus reducing the amount of state aid needed by poor people who were too old to work.102 Maseres’s procedures are prosaic but unmistakably algorithmic:

借助斯马特先生的第二张复利表或其他方法，根据给定的利率，计算给定年数后可收到的某1英镑的现值。然后在给定的人类寿命在各个不同年龄段的概率表中，找出与购买者年龄相当的人数。然后将购买者的年龄加上到期应收到1英镑款项的年数，并在该表中找出与上述较大年龄相当的人数。103

Find the present value of one pound certain, to be received at the end of the given number of years, according to the given rate of interest, by the help of Mr. Smart’s second table of compound interest, or otherwise. Then find in the given table of the probabilities of the duration of human life at the several different ages of it, the number of persons living at the age of the purchaser. Then add to the age of the purchaser the number of years at the end of which the sum of one pound is made payable to him, and find in the said table the number of persons living at the said greater age.103

等等。本书还通过象征性和例子介绍了同样的程序，以便使其“尽可能清晰和熟悉”，马塞雷斯写道。104

And so forth. The book also presents the same procedure symbolically and through examples so as to make it, Maseres writes, “as clear and familiar as possible.”104

这段话枯燥乏味，很容易让人忽视它是关于预测死亡的事实。105由于年金支付的是人的一生，发行人的最终成本取决于购买者的寿命，因此发行人可以根据预期寿命信息调整价格，从而将风险降至最低（图3.2）。虽然死亡率表早已为政府目的编制，但将其用于设定费率却是新鲜事：第一家使用它们的人寿保险公司成立于 1762 年。106如何最好地利用这些统计数据是 Lorraine Daston 所说的“古典概率论”的核心问题。107孔多塞是这个学派的主要代表，它与现代概率论的不同之处在于，它没有明确区分客观概率和主观信念程度。该学派的后期信徒之一皮埃尔-西蒙·拉普拉斯有一句名言，概率论不过是“简化为微积分的良好判断”；目标是将人类思维中推理偶然事件的先天能力形式化。108

The dryness of this passage can easily lead one to overlook the fact that it is about predicting death.105 Since an annuity pays out for a person’s entire life, the eventual cost to the issuer depends on how long the buyer lives, and so the issuer could minimize risk by adjusting the price based on information about life expectancy (figure 3.2). While mortality tables had long been compiled for governmental purposes, their use to set rates was new: the first life insurance company to use them was founded in 1762.106 How best to use such statistics was a central problem of what Lorraine Daston has called “the classical theory of probability.”107 This school of thought, of which Condorcet was a major exponent, differed from modern probability theory in that it drew no clear distinction between objective probabilities and subjective degrees of belief. One of the later adherents of this school, Pierre-Simon Laplace, famously stated that probability theory was nothing but “good sense reduced to a calculus”; the goal was to formalize the human mind’s innate ability to reason about matters of chance.108

与孔多塞一样，马塞雷斯也声称自己的理论建立在常识之上，并得到了常识的认可。但马塞雷斯的方法为比他同时代的法国理论更重视个人能动性。虽然他坚称自己的数据可靠，但他也拒绝将计算结果作为最终结论：必须运用“判断力和自由裁量权”来决定是否应在特定情况下例外。109同样，在一本关于年金法案的小册子中，他建议允许“如果教会委员或济贫监督员认为合适”，可以偏离规定的利率。110从数学上讲，这种立场转化为一种坚持致力于使技术定义与人们（在他看来）通常的思维方式保持一致。他信守承诺，在年金书中严格避免使用负面因素；即使在表格中，他也始终将差异视为正面因素，无论其方向如何。他还对死亡率表和计算方法进行了广泛的比较，并根据其概念的合理性以及他认为的结果的合理性进行评判。

Like Condorcet, Maseres presented his theory as grounded on and legitimized by common sense. But Maseres’s approach left more room for individual agency than the theories of his French contemporaries. While he maintains that his numbers are solid, he also declines to take the calculations as the final word: one must use “judgment and discretion” in determining whether exceptions should be made in particular cases.109 Likewise, in a pamphlet on the annuity bill, he suggests allowing for deviations from the prescribed interest rate “if the Churchwardens or Overseers of the poor shall think fit.”110 Mathematically, this stance translated into an insistence on keeping technical definitions in line with the ways people (in his view) ordinarily think. True to his word, he scrupulously avoids negatives in the book on annuities; even in tables, he always presents differences as positive regardless of their direction. He also undertakes extensive comparisons of mortality tables and computational methods, which he judges based on both their conceptual soundness and what he takes to be the reasonableness of their results.

对于 21 世纪的算法能力批评者来说，这似乎是一种令人耳目一新的人文主义方法。与一些批判算法研究的潮流非常相似，马塞雷斯反对盲目信任规则。然而，他的常识性方法也有其局限性。马塞雷斯最具煽动性的追随者威廉·弗伦德的作品中，政治倾向变得更加明显。弗伦德因在 1793 年撰写了一本名为《向共和党和反共和党联合组织推荐的和平与联盟》的小册子而臭名昭著，其中包含对导致他被剑桥开除的神职人员的批评。他 1796 年的著作《代数原理》包含马塞雷斯的长篇附录，试图从常识的角度对代数进行全面的论述。在《和平与联盟》中，他强调制定被统治者能够理解的法律的重要性；他的代数同样应该排除那些无法理解因而不得不相信的想法（例如，否定）。111这一论点的反天主教色彩显而易见：弗伦德将否定词比作《亚他那修信经》，并与蒂洛森大主教一起宣称：“我希望我们彻底摆脱它！” 112

To twenty-first-century critics of algorithmic power, this may seem a refreshingly humanistic approach. Much like some currents of critical algorithm studies, Maseres pushes against blind trust in rules. Yet his commonsense approach has its limitations. The political thrust became more overt in the work of Maseres’s most incendiary follower, William Frend. Frend was notorious for writing a 1793 pamphlet entitled Peace and Union Recommended to the Associated Bodies of Republicans and Anti-Republicans, which contained criticisms of the clergy that got him expelled from Cambridge. His 1796 book Principles of Algebra, which includes a lengthy appendix by Maseres, attempts a comprehensive treatment of algebra on commonsense grounds. In Peace and Union, he stresses the importance of writing laws that the governed can understand; his algebra was similarly supposed to exclude ideas (e.g., negatives) that were incomprehensible and that thus had to be taken on trust.111 The anti-Catholic overtones of this argument are apparent: Frend likens negatives to the Athanasian Creed, declaring with Archbishop Tillotson, “I wish we were fairly rid of it!”112

弗伦德的代数教材强调了语言在十八世纪关于数的本质的争论中的核心地位。他告诉我们，他的目标是用一种适合“英国男孩和女孩”的语言来解释代数——也就是说，用一种直接的、非隐喻的、地道的英语，避免使用诸如二次方和平方数之类的“外来”或专业术语。113他写道：“人们常常错误地认为，一个词即使解释得通，也无足轻重。如果这个词在其他方面含义截然不同，学习者就会经常混淆这些不同的含义，最终对这个主题一无所知。” 114也就是说，即使赋予“平方”一词一个全新的专业定义，它所蕴含的联想也将不可避免地影响人们对平方数的理解。在十八世纪代数的语境中，这种关于词语对思维影响的立场不仅在教学法上，而且在认识论上也具有深远的影响。例如，要求遵循公认的惯例会阻碍类似欧拉对“和”重新定义的论证。对欧拉和孔多塞来说，数学家可以自由地定义他们喜欢的词语，因此明确的定义可以解决争议。而对弗伦德来说，这样的定义不值得信任，因为它们可能永远不会完全取代词语的传统含义。正如米凯利斯宣称语言是一种民主，由人民而非学者统治，弗伦德在其国家的日常词汇中寻求真正的概念。

Frend’s algebra text underscores the centrality of language to the eighteenth-century debates about the nature of number. His goal, he tells us, is to explain algebra in a language suitable for “English boys and girls”—that is, direct, nonmetaphorical, idiomatic English that avoids “foreign” or technical terms such as quadratic and square number.113 “People err much,” he writes, “in supposing that a word is of little consequence, if it is explained. If that word has a very different meaning in other respects, the learner will confound frequently the different meanings, and pass through life without having a clear idea upon the subject.”114 That is, the associations already attached to the word square will inevitably shape how one understands square numbers even if the word is given a wholly new technical definition. In the context of eighteenth-century algebra, this position about the influence of words on thought had epistemological, not just pedagogical, ramifications. Demanding adherence to received usage would, for instance, preclude argumentation along the lines of Euler’s redefinition of sum. For Euler and Condorcet, mathematicians could freely define words however they pleased, and explicit definitions could thus settle controversy. For Frend, such definitions were to be distrusted since they could never wholly displace the conventional meanings of words. As Michaelis had declared language a democracy, governed by the people rather than by the learned, Frend sought true notions in the everyday vocabulary of his country.

然而，马塞雷斯和弗伦德推动的更多的是渐进式改革，而非完全民主的社会。弗伦德在《和平与联盟》中指出，制定更宽容的法律有助于“保持对既定权威更稳定的依恋”。115弗伦德可能想用这个论点来挑衅，但它恰恰描述了马塞雷斯担任魁北克总检察长期间所采取的做法。魁北克是一个以天主教徒和法语为主的殖民地，英国在 1759 年七年战争期间从法国手中夺取了它。在伯克等人的建议下，英国人在魁北克建立了宗教宽容，作为一种务实的措施，因为取缔天主教会导致其秘密活动。116马塞雷斯在处理土地权问题上也采取了类似的宽容策略。1767 年，他参与了一项财产法的努力，该法允许该国既定做法保持一定程度的连续性，从而避免疏远法裔加拿大民众。这种方法与伯克不久后批评的“几何”方案形成鲜明对比。马塞雷斯起草的法律草案指出，为了避免“扰乱人们古老而习以为常的权利以及以此为基础的自然期望”，法国关于土地权的习俗“被认为自征服该国之时起就一直延续，从未间断”。117在这种情况下，延续性和传统起到了保护作用，避免了突然变化可能造成的混乱和不满，从而最终成为巩固英国统治的一种方式。

And yet Maseres and Frend were pushing more for incremental reforms than for a fully democratic society. Making laws more tolerant, Frend argues in Peace and Union, helps “preserve a steadier attachment to the established authority.”115 Frend may have meant this argument as a rhetorical wedge, but it describes just the approach Maseres took while serving as attorney general of Quebec. Quebec was a predominantly Catholic and Francophone colony that Britain had captured from France in 1759, during the Seven Years’ War. On the advice of Burke and others, the British established religious tolerance in Quebec as a pragmatic measure because outlawing Catholicism would have led to its clandestine practice.116 Maseres took a similarly tolerant tack in dealing with land rights. In 1767, he was involved in an effort to establish property laws that would allow for some degree of continuity in the established practices of the country so as to avoid alienating the French Canadian populace. This approach contrasts with the “geometrical” scheme that Burke would soon criticize. As a way to avoid “unsettling men’s ancient and accustomed rights and natural expectations founded thereon,” states the draft law that Maseres prepared, French customs regarding land rights are “deemed and taken to have continued without interruption from the time of the conquest of this country.”117 Continuity and tradition serve, in this instance, as a protection against the confusion and discontent that sudden change would cause and thus, ultimately, as a way of shoring up British rule.

因此，马塞雷斯更倾向于“判断和自由裁量权”而非严格的规则，这并不一定会将权力置于人民手中。将一个系统与一个国家的现行实践相结合，反而会巩固制定规则者的权威。在其关于年金的著作中，马塞雷斯认为规则的必要性主要在于保护平民免遭自身侵害：法律必须保护购买年金的穷人“免受自身愚蠢和软弱的侵害，使他们无法在未经适当深思熟虑的情况下，以低于其真实价值的一小部分的价格出售年金” 。118这一论点表明，马塞雷斯所信任的判断似乎主要来自像他这样的绅士，而不是下层阶级。鉴于他在财政法院的地位，这种立场尤其会引起共鸣。当时，财政部的运作几乎不受议会的监督，改革者认为，像马塞雷这样的职位是作为政治恩惠而发放的、不必要的高薪闲职。119在这些批评者眼中，财政部是一个不透明的机构，按照其自私自利的传统运作——简而言之，正是法国大革命威胁要根除的那种根深蒂固的利益。

Maseres’s preference for “judgement and discretion” over strict rules, then, does not necessarily place power in the hands of the people. Fitting a system to the existing practices of a nation can instead solidify the authority of those who write the rules. In his book on annuities, Maseres treats rules as necessary primarily to protect commoners from themselves: the law must secure the poor who buy annuities “against their own folly and weakness, by making it impossible for them to sell their annuities for a small part of their true value, over a pot of ale and without a proper degree of deliberation.”118 The judgment Maseres trusted, this argument suggests, would seem to be primarily that of gentlemen like himself, not that of the lower classes. This stance would have had a particular resonance given his position on the Court of the Exchequer. The Exchequer operated with little oversight from Parliament at the time, and reformers were arguing that positions like the one Maseres held were needlessly high-paying sinecures given out as political favors.119 In the eyes of such critics, the Exchequer was an opaque institution that operated by its own self-serving traditions—in short, just the sort of entrenched interest the French Revolution was threatening to uproot.

孔多塞与马塞雷之间的差异——前者热衷于消除语言，后者热衷于保留语言——或许部分源于民族背景。18 世纪，欧洲大陆的数学家采用了莱布尼茨的微积分符号，而英国数学家则大多追随牛顿，厌恶过于依赖符号的方法。普遍的说法是，这种厌恶阻碍了英国数学家的发展，直到 19 世纪 10 年代，查尔斯·巴贝奇等人通过翻译西尔维斯特-弗朗索瓦·拉克鲁瓦的微积分教科书，将法国数学思想引入剑桥。120但民族传统的区分并不像人们通常认为的那样泾渭分明。18 世纪后期，英国出现了一种与孔多塞类似的轰动性认识论。这种思想的拥护者之一是孔多塞的英国盟友查尔斯·马洪，斯坦霍普伯爵第三代。对他来说，数学理性是对抗财政部所代表的自给自足特权的武器。斯坦霍普勋爵的例子表明，当时不同的认识论并非完全归因于民族文化——它们也具有政治意义。

The difference between Condorcet and Maseres—the former keen to eliminate language, the latter keen to keep it in sight—may be ascribed in part to national context. In the eighteenth century, Continental mathematicians adopted Leibniz’s notation for calculus, whereas the English largely followed Newton in his aversion to methods that depended too strongly on symbols. The received story is that this aversion held English mathematicians back until, in the 1810s, Charles Babbage and others introduced French mathematical ideas to Cambridge through their translation of Silvestre-François Lacroix’s textbook on calculus.120 But the differentiation of national traditions is not as clear-cut as often supposed. Sensationalist epistemology not unlike that of Condorcet was present in England in the late eighteenth century. Among the champions of such thinking was Condorcet’s English ally Charles Mahon, third Earl of Stanhope, for whom mathematical reason served as a weapon against the self-sustaining privilege the Exchequer represented. The example of Lord Stanhope shows that the divergent epistemologies of the period are not wholly reducible to national culture—they also had a political meaning.

理性时代的逻辑机器

Logic Machines in the Age of Reason

孔多塞侯爵写过关于机械化逻辑推理的文章；斯坦霍普实际上做到了。《哲学年鉴》 1818年一篇关于这位伯爵的匿名讣告解释道：“据权威人士断言，他设想了制造一种推理机器的可能性，通过这种机器，可以像确定数字结果一样轻松准确地确定某些想法组合或基本命题的结果。” 121斯坦霍普并没有建造我们今天所理解的推理机器。然而，他确实构建了马丁·加德纳所说的逻辑机：“一种专门为解决形式逻辑问题而设计的电气或机械装置。” 122当时逻辑学是一门与代数完全不同的学科，因此这些装置不能归入莱布尼茨和孔多塞的通用代数传统。然而，斯坦霍普的机器——他大约在1801年开始建造，基于他数十年来一直研究的理论——也表达了同样的愿望，即通过用更清晰的语言取代语言来让人们达成共识。斯坦霍普写道，“演示器”将“通过一个纯粹机械的符号，而不是使用任何符号”来展示推理。这就是所谓的词语。” 123这样，机器就能打破言语辩论的诡辩，揭示无人能否认的真理。

The Marquis de Condorcet wrote about mechanizing logical reasoning; Stanhope actually did it. An anonymous 1818 obituary of the earl in Annals of Philosophy explains: “It has been asserted, upon grave authority, that his Lordship conceived the possibility of forming a reasoning machine, by which the results of certain combinations of ideas, or of elementary propositions, might be ascertained with as much ease and accuracy as those of figures.”121 Stanhope did not build a reasoning machine as we might understand the term today. He did, however, construct what Martin Gardner calls a logic machine: “a device, electrical or mechanical, designed specifically for solving problems in formal logic.”122 Logic was an entirely separate discipline from algebra at the time, and so these devices cannot be placed in the universal-algebra tradition of Leibniz and Condorcet. Yet Stanhope’s machines, which he began building around 1801 based on a theory he had been working on for decades, express much the same desire to bring people into agreement by replacing language with something clearer. The “Demonstrator” will display reasoning, Stanhope wrote, “by means of a symbol purely mechanical, and without using any of those symbols which are called words.”123 The machine would thus cut through the sophistry of verbal argumentation to reveal truths that no one could deny.

只有结合斯坦霍普激进的民主政治立场，才能理解他建造“推理机器”的动机。他是孔多塞的密友，两人就一系列数学和政治问题进行交流。这位被同时代人称为“雅各宾伯爵”的人物还与弗朗索瓦-亚历山大-弗雷德里克·德·拉罗什富科及其他法国大革命人物有私交，即使在法国向英国宣战后，他仍继续支持他们。124不出所料，这一立场并未为他在上议院赢得多少盟友，斯坦霍普自己的子女也对他咄咄逼人的观点感到愤怒。他的儿子二十岁离家出走，女儿海丝特·斯坦霍普夫人与他断绝关系，自称“贵族”，并谴责他与之交往的“肮脏的雅各宾派”。125在经历了一系列失望之后，伯爵将时间投入到科学和工程实践中。和他经常推崇的托马斯·潘恩的作品一样，他提倡用科学理性来挑战贵族的权威。126演示器是他众多发明之一，其他发明还包括一系列机械计算器、一种建筑物防火装置，以及一种在 19 世纪初被广泛采用的新型印刷机。

Stanhope’s reasons for building “reasoning machines” can only be understood in light of his radically democratic politics. He was a close friend of Condorcet, and the two corresponded on a range of mathematical and political matters. The “Jacobin Earl,” as his contemporaries called him, also had personal ties to François-Alexandre-Frédéric de La Rochefoucauld and other figures involved in the French Revolution, and he continued to support them even after France declared war on Britain.124 Predictably, this position did not win him many allies in the House of Lords, and Stanhope’s own children bristled at his leveling views. His son ran away from home at the age of twenty, and his daughter, Lady Hester Stanhope, cut ties with him, declaring herself “an aristocrat” and denouncing the “dirty Jacobins” with which he associated.125 Through this whole series of disappointments, the earl devoted his time to practical efforts in science and engineering. Like Thomas Paine, whose work he often championed, he advocated scientific reason as a way of challenging the authority of the aristocracy.126 The Demonstrator was one of his numerous inventions, which also included a series of mechanical calculators, a means of fireproofing buildings, and a new type of printing press that was widely adopted in the early nineteenth century.

从哲学上讲，斯坦霍普的逻辑体系与孔狄亚克神父的后期作品相一致。127作为那个时代的特征，斯坦霍普关注如何使逻辑“富有成效”——也就是说，让它能够产生关于世界的新知识，而不是仅仅以不同的形式重申前提。斯坦霍普的成年生活从 18 世纪 70 年代延续到 19 世纪 10 年代，大致相当于威尔伯·塞缪尔·豪厄尔所说的“新逻辑”的巅峰时期。128当兰伯特和戈特弗里德·普卢凯等莱布尼茨主义者尝试用数学方法研究逻辑时，另一个逻辑思想流派，包括孔狄亚克以及托马斯·里德和乔治·坎贝尔，则避开形式主义，转而支持实践教育。这个学派通常对亚里士多德三段论的实用性持怀疑态度；坎贝尔在《修辞哲学》 （1776 年）中宣称，没有人“能够成为需要推理的人”。129与旧的基于规则的体系相比，“新逻辑”旨在训练人们在现实生活中清晰、有条理、不带偏见地思考。

Philosophically, Stanhope’s logic system was aligned with the later work of the Abbé de Condillac.127 Characteristically of the period, Stanhope shows a concern with making logic “productive”—that is, enabling it to produce new knowledge about the world rather than merely restating premises in different forms. Stanhope’s adult life, stretching from the 1770s to the 1810s, roughly corresponds to the peak of what Wilbur Samuel Howell has called “the new logic.”128 While Leibnizians such as Lambert and Gottfried Ploucquet were experimenting with mathematical approaches to logic, another school of logical thought, including Condillac as well as Thomas Reid and George Campbell, shunned formalism in favor of practical education. This school was generally skeptical of the utility of Aristotelian syllogisms; in The Philosophy of Rhetoric (1776), Campbell declares that no one “will ever be made a reasoner, who stands in need of them.”129 In contrast to the old rule-based system, the “new logic” was meant to train people to think clearly, methodically, and without prejudice in real-life situations.

斯坦霍普的逻辑理论在某种程度上试图将三段论从这种批评中拯救出来，通过证明它们可以产生不仅仅是重言式的论证。其基本思想是将数值量引入范畴论逻辑。我们无法从“有些人很穷”和“有些人很聪明”这两个前提中逻辑地推断出任何东西，但如果有“60%的人很穷”和“60%的人很聪明”，我们就可以推断出至少20%的穷人是聪明的。斯坦霍普的装置以可视化的方式执行此类推断（图3.3）。它使用方形孔径表示整个类别，斯坦霍普称之为全息图（holos）或全子图（holon）；两个滑块穿过插槽，以表示构成该整体一定比例的其他类别。如果强制滑块重叠，就可以推断出它们之间的关系。以罗伯特·哈利为例，假设我们有前提“没有吹牛的人值得尊重”和“有些英雄是吹牛的人” 。130我们用全子来表示吹牛的人，因为这是这两个前提共同的术语。一个滑块代表不值得尊重的人；由于这包括所有吹牛的人，因此滑块必须覆盖整个孔径。另一个滑块代表英雄；由于可能还有其他不是英雄的吹牛的人，因此这个滑块只覆盖部分孔径。因为滑块必须重叠，所以我们可以得出结论“有些英雄不值得尊重”。

Stanhope’s logical theory was in part an attempt to rescue syllogisms from this criticism by showing that they could produce more than mere tautologies. The basic idea is to introduce numerical quantities into categorical logic. One cannot logically infer anything from the premises “Some people are poor” and “Some people are intelligent,” but if one has “Sixty percent of people are poor” and “Sixty percent of people are intelligent,” one can infer that at least twenty percent of the poor are intelligent. Stanhope’s device performs such inferences visually (figure 3.3). It represents a whole category using a square aperture, which Stanhope referred to as the holos or holon; two sliders are inserted through slots so as to represent other categories that make up some proportion of this whole. If the sliders are forced to overlap, a conclusion can be deduced about their relation. To take an example from Robert Harley, suppose we have the premises “No boaster deserves respect” and “Some heroes are boasters.”130 We use the holon to represent boasters, since this is the term that is common to the two premises. One slider represents people who do not deserve respect; since this includes all boasters, the slider must cover the whole aperture. The other slider represents heroes; since there may be other boasters who are not heroes, this slider covers only some of the aperture. Because the sliders must overlap, we can conclude that “Some heroes do not deserve respect.”

这个想法似乎与我们如今理解的符号计算关系不大。当然，斯坦霍普的体系缺乏莱布尼茨和布尔逻辑体系所特有的代数基础。但《论证者》提出的关于符号的认识论问题与这些思想家面临的问题并无二致。1818年的讣告对逻辑论证者能否获得真正的确定性并不乐观：

This idea may not seem to have much to do with symbolic computation as we now understand it. Certainly, Stanhope’s system lacks the grounding in algebra that characterizes the logical systems of Leibniz and Boole. But the Demonstrator raised epistemological questions about symbols not unlike the ones those thinkers faced. The 1818 obituary was not sanguine about the idea that a logic demonstrator could achieve true certainty:

几乎无需赘述，撇开其他困难不谈，任何机械的推理过程都无法运用，除非人类对某些普遍原则达成了明确的共识，就像对某些数字的价值达成共识一样，除非所有关于词语含义及其组合的疑问都已消除。一台解决政治问题的机器，其答案会大相径庭，因为它是由一位改革倡导者监督建造的，还是由一位我们祖先绝对智慧的崇拜者监督建造的。131

It is scarcely necessary to observe that, independent of other difficulties, no mechanical process for reasoning can ever be employed until mankind have agreed upon certain general principles as decidedly as upon the value of certain numbers, and until all doubt has been removed respecting the import of words, or the combinations of them. A machine for resolving political queries would give very different answers, according as it was constructed under the superintendance of an advocate for reform, or an admirer of the infallible wisdom of our ancestors.131

对斯坦霍普项目的批评，可以说是二十一世纪一种常见批评观点的先兆：机器反映了其创造者的政治偏见。斯坦霍普与算法公平的倡导者一样，正在为那些本应属于修辞学领域的问题寻找技术解决方案。这位讣告作者反驳道，这种中立立场不过是掩盖其意图的花招。

This critique of Stanhope’s project is arguably a precursor of what has become a critical commonplace in the twenty-first century: machines reflect the political biases of their creators. Stanhope, like advocates of algorithmic fairness, is looking for technical solutions for problems that properly belong to the domain of rhetoric. This stance of neutrality, the obituary writer counters, is sleight of hand concealing an agenda.

但斯坦霍普的中立性并非21世纪技术官僚赋予算法的那种中立性。正如经济史学家希洛夫和西兰蒂耶夫所观察到的，使用演示器所需的工作量与自己思考三段论所需的工作量大致相同。132这正是关键所在。斯坦霍普认为逻辑的机械化主要是为了“增强人类思维”和提供“对抗自负的良药”；正如一位同时代的传记作家所说，机器的目的“不仅在于检测错误的推理，无论它们组合得多么复杂，还在于揭示推导出这些错误结论的链条的各个环节”。133这种对揭示推理链条的坚持，我们可以将其比作莱布尼茨在《艺术》中描述的“试验或测试”。“发现”一词在斯坦霍普的语境中具有政治意味。在《政治正义论》（1793 年）中，威廉·戈德温坚持认为人们必须了解事物背后的原因：“如果真理存在于头脑中，而没有它所依赖的证据，那么就不能说是被理解了。” 134虽然戈德温在这里指的是“政府假定将我们从独立思考的麻烦中解救出来”的情况，但他的论点同样适用于推理机器的设计。135机器不应该为你思考；它的目的是让你思考。

But Stanhope’s neutrality is not the neutrality twenty-first-century technocrats assign to algorithms. As the economic historians Shilov and Silantiev observe, using the Demonstrator requires about as much work as it would take to think through the syllogism oneself.132 This difficulty is precisely the point. Stanhope viewed the mechanization of logic primarily as a way to “strengthen the human mind” and provide “an anti-dote to self-conceit”; as a contemporaneous biographer puts it, the machine is meant “not only to detect false reasoning, however sophistically combined, but to shew the various links of the chain by which these false conclusions have been deduced.”133 This insistence on revealing chains of reasoning, which we might liken to the “trials or tests” Leibniz describes in “The Art of Discovery,” had a political thrust in Stanhope’s context. In An Enquiry Concerning Political Justice (1793), William Godwin insisted that one must know the reasons behind things: “Wherever truth stands in the mind unaccompanied by the evidence on which it depends, it cannot properly be said to be apprehended at all.”134 Although Godwin is here referring to the situation in which “government assumes to deliver us from the trouble of thinking for ourselves,” his argument could just as well apply to the design of a reasoning machine.135 The machine is not supposed to think for you; its purpose is to make you think.

这种解释仍然存在一个难题。如果结果是基于将自然理性应用于人人皆可获取的感官数据，那为什么还需要机器呢？为什么不像戈德温所写的那样，独立思考呢？斯坦霍普试图实现的目标或许可以从他的政治生涯中找到线索。作为英国议会议员，他推动将理性原则编入可执行的法律中，以此来防止腐败。例如，1786年，他出版了一本小册子，批评首相小威廉·皮特削减国债的计划。皮特的计划包括设立一个委员会，负责赎回他们“认为适宜”赎回的任何年金。136（值得注意的是，其中一位委员是马塞雷斯的上司，财政大臣。）斯坦霍普在他的小册子中指出，皮特的计划将使委员们能够“通过赌博公共资金”发财——也就是说，这将使他们能够私下购买股票，因为他们知道，随着政府资金的注入，股票的价值很快就会上升。137

This interpretation still leaves a difficulty. If the results are based on the application of natural reason to sensory data that are accessible to everyone, why is the machine needed at all? Why not just, as Godwin writes, think for oneself? A clue as to what Stanhope was trying to achieve may be found in his political career. As a member of British Parliament, he pushed for encoding rational principles in enforceable laws as a way of preventing corruption. In 1786, for instance, he published a pamphlet criticizing the plan of the prime minister, William Pitt the Younger, for reducing the national debt. Pitt’s plan involved creating a commission that could redeem any annuities they “deem it expedient” to redeem.136 (One of the commissioners, it is worth noting, was to be Maseres’s boss, the chancellor of the Exchequer.) Pitt’s plan, Stanhope argued in his pamphlet, would enable the commissioners to make fortunes “by gambling in the public funds”—that is, it would enable them to buy up stocks personally, knowing that their values will soon be increased by an infusion of government money.137

这一论点在议会中并未得到很好的反响。在随后的辩论中，巴瑟斯特伯爵认为斯坦霍普提到赌博是对委员荣誉的侮辱。138但斯坦霍普在辩论中表示，他担心皮特的计划取决于未来管理者的善意，而他们可能并不那么值得信赖。他的替代计划将使这笔资金“不可转让”，这样大臣们就永远不能将其挪用于个人或政治利益。139众所周知，斯坦霍普以一种模仿数学证明结构的风格发表了这个计划，以“公理 I ”开头，以类似 QED 的声明结尾：“这是我提议证明的命题”；根据后来的记载，他甚至使用了他的一台计算机来确保他的数值结果的正确性。140事实上，斯坦霍普的证明并没有令人信服；皮特的法案获得通过。

This argument did not go over especially well in Parliament. In the ensuing debate, the Earl of Bathurst took Stanhope’s reference to gambling as a slight against the commissioners’ honor.138 But Stanhope was concerned, as he stated in the debate, that Pitt’s plan depended on the goodwill of future administrators, who may not be so trustworthy. His alternative plan would render the fund “UNALIENABLE” so that ministers can never divert it for personal or political gain.139 Notoriously, Stanhope published this plan in a style that mimics the structure of a mathematical proof, beginning with “AXIOM I” and ending with the QED-like pronouncement, “And this is the proposition, which I had proposed to prove”; he even, according to a later account, employed one of his calculating machines to ensure the correctness of his numerical results.140 As it happened, Stanhope’s proof did not persuade; Pitt’s bill passed.

这场争论的焦点再次在于，谁的判断值得信任。皮特的计划将如何处理资金的决定权交给了指定人员，而该计划的实施也依赖于这些人员的信誉。斯坦霍普的回应体现了卢梭对人性的描绘，这种描绘体现了十八世纪后期出现了如此多的政治激进主义。人性本善，理性本性，但实际上他们已被社会力量腐蚀，因此无法指望他们公正地推理。马塞雷斯认为平民身上存在这种腐败，而斯坦霍普则尤其不信任贵族，因为他们的社会归属和政治承诺会促使他们歪曲事实。141在这方面，“示威者”的作用与法治之于托马斯·潘恩的作用大致相同——它确保任何人，即使是当权者，都不能凌驾于理性法则之上。如果说斯坦霍普的作品中有什么方面预示了现代算法思维，那就是他对精心策划的系统而非非正式协议和个人自由裁量权的偏好。与法国大革命中的某些力量一样，斯坦霍普看到了规则的积极价值，它承诺通过确保决策的一致性来根除既得利益。142

At issue in this debate was, once again, whose judgment was trusted. Pitt’s plan left decisions about how to handle funds to the discretion of appointed individuals, on whose honor the plan relied. Stanhope’s response is characteristic of the Rousseauian picture of human nature that informed so much political radicalism in the late eighteenth century. People are naturally rational and naturally good, but in practice they have been corrupted by social forces, and so they cannot be relied on to reason justly. Whereas Maseres perceived this corruption in commoners, Stanhope specifically distrusted aristocrats since their social attachments and political commitments would give them an impetus to distort the truth.141 In this regard, the Demonstrator would have served much the same function that the rule of law served for Thomas Paine—it would ensure that no one, not even those in power, stood above the laws of reason. If there is an aspect of Stanhope’s work that anticipates modern algorithmic thinking, it is this preference for intentionally planned systems over informal agreements and individual discretion. Like some forces within the French Revolution, Stanhope saw a positive value in rules, which promised a way of rooting out entrenched interests by ensuring that decisions were made consistently.142

然而，斯坦霍普想要构建的系统并非现代意义上的算法。与艾伦·图灵的“有效程序”不同，演示器的价值评判标准并非仅仅在于它能否高效地生成正确的输出；相反，它应该能够帮助人们理性地解决彼此之间的差异。为了实现这一政治目标，该系统不能依赖于需要专门训练才能理解的技术术语。斯坦霍普与孔多塞有着共同的渴望。这种渴望在马塞雷斯身上也体现出来，他希望将代数植根于日常语言中，以确保其易于理解。孔多塞和斯坦霍普与马塞雷斯和弗伦德这两派之间的分歧在于如何建立这种共同点以及将哪些人纳入其中。马塞雷斯将理性置于绅士们现有的常识中，并信赖其民族现有的语言。斯坦霍普致力于创造一种新的、合乎逻辑的交流方式，他希望将每个人都纳入其中。

And yet the systems Stanhope wanted to construct were not exactly algorithms in the modern sense. Unlike Alan Turing’s “effective procedures,” the Demonstrator was not to be judged solely by how efficiently it could produce the correct output; instead, it was supposed to enable people to work out their differences rationally. To fulfill this political purpose, the system could not depend on a technical cant that required specialized training to understand. Stanhope shared this desire for a common ground with Condorcet. This desire may also be discerned in Maseres, who wanted to root algebra in everyday language as a way to ensure that it is comprehensible. The difference between the two factions—Condorcet and Stanhope on one side and Maseres and Frend on the other—regarded how this common ground would be established and who would be included. Maseres located reason in the existing common sense of gentlemen and trusted in the existing language of his nation. Stanhope was out to create a new, logical form of communication, and he wanted to include everyone.

这项平等计划注定无法实现。如果说马塞雷斯对既定做法过于乐观，那么斯坦霍普和孔多塞则对取代这些做法的容易程度过于乐观。双方都没有赢。到1801年，当斯坦霍普开始创作《论证者》时，他所坚持的认识论已经走向衰落；在接下来的一个世纪里，数学家和逻辑学家放弃了社会改革的雄心，数学与常识之间的鸿沟变得难以逾越。当然，通用代数的梦想并没有消失，但它呈现出一种新的形式，在这种形式中，将计算的机械操作与人类的实际思维结合起来不再显得那么重要，而人们所认为的词语的背信弃义也不再显得那么成问题。不同的语言观之间的冲突让位于一种妥协：符号与文字可以协同作用，符号提供严谨性，文字提供意义，这构成了人类思想培养的重要组成部分。要理解这种妥协如何实现，有必要探讨1790年代与激进主义和传统主义派系之间存在复杂关系的另一种思想潮流：浪漫主义。

This leveling program was not destined to be realized. If Maseres was too sanguine about received practices, Stanhope and Condorcet were too optimistic about the ease of replacing them. Neither side won. By 1801, when he began working on the Demonstrators, Stanhope was clinging to an epistemology that was already on its way out; in the ensuing century, mathematicians and logicians withdrew their ambitions of social reform, and the gap between mathematics and common sense became insurmountable. The dream of a universal algebra did not, to be sure, disappear, but it took on a new form in which aligning the mechanical operations of computation with actual human thought no longer seemed so important, and in which the perceived perfidy of words no longer seemed like such a problem. The clashes between disparate views of language gave way to a compromise: symbols and words could work together, with the symbols providing rigor and the words providing the meaning that formed an essential component of the cultivation of human thought. To understand how this compromise was possible, it is necessary to consider another intellectual current that existed in a complex relation to the radical and traditionalist factions in the 1790s: Romanticism.

脱离光明的统治

Out of the Light’s Dominion

启蒙运动对文化存在质疑。当启蒙思想家们讨论人们从前几代人那里继承下来的观念时，他们大多用负面词汇来描述它们，称其为“错误”和“偏见”，为了传播光明，必须将其消除。孔多塞认为，政府一再犯错，他们“将偏见和恶习转化为积极的用途，而不是试图消除或压制它们” 。143他进一步阐述道，这种做法源于“错误地将自然人等同于现有文明状态的产物，即被偏见、人为激情和社会习俗腐蚀的人”。144虽然孔多塞有时使用“偏见”（法语为préjugé ）一词来指代针对特定群体的偏见，但它在18世纪具有更广泛的含义。145在《百科全书》中，路易·德·若库尔将偏见定义为“在智力运作不充分的情况下，灵魂对事物本质做出的错误判断”。146人们通常认为这些错误判断是通过语言传播的，因此语言可以成为人们对无根据的先入之见、学术错误和错误的民间信仰的延续的更广泛担忧的替罪羊。博学者皮埃尔·路易·莫佩尔蒂写道：“我们几乎还没出生，就听到无数重复的词语表达了我们周围人的偏见，而不是精神中诞生的最初想法。” 147孔多塞版本的启蒙运动意味着通过用更好的东西取代语言来打破偏见代代相传的传递链。

The Enlightenment had a problem with culture. When they discussed the conceptions people receive from past generations, the philosophes largely cast them in negative terms, as “errors” and “prejudices” that had to be expunged in the interest of spreading light. According to Condorcet, governments have repeatedly erred in the practice “of turning prejudices and vices to good account rather than trying to dispel or repress them.”143 This practice, he elaborates, stems from the “mistake of identifying the natural man with the product of the existing state of civilization, with, that is, man corrupted by prejudices, artificial passions and social customs.”144 While Condorcet sometimes used the word prejudice—in French, préjugé—to refer to biases against groups of people, it had a broader meaning in the eighteenth century.145 In the Encyclopédie, Louis de Jaucourt defines prejudice as a “false judgement that the soul carries of the nature of things, after an insufficient exercise of the intellectual faculties.”146 It was often assumed that these false judgments were transmitted through language, which could therefore serve as scapegoat for broader concerns about the perpetuation of unwarranted preconceptions, scholastic errors, and mistaken folk beliefs. “Hardly are we born,” wrote the polymath Pierre Louis Maupertuis, “but we hear repeating an infinity of words that express rather the prejudices of those who surround us than the first ideas that are born in the spirit.”147 Condorcet’s version of Enlightenment meant breaking the chain of transmission by which prejudices persist across generations by replacing language with something better.

这种对文化的反感也有其弊端。由于目标是重新开始，既有的实践，尤其是语言，似乎对科学构成了威胁。孔多塞希望，即使所有语言都消失了，他的通用代数也能被理解；就像斯瓦尔巴群岛的知识种子库一样，它可以提供一种保障，避免陷入黑暗和混乱，就像他担心法国在恐怖统治时期所面临的那样。148 他实现这种普遍可理解性的方法以洛克式的煽情主义为蓝本，他写道，这种煽情主义“永远在人类与其婴儿期的错误之间筑起了一道屏障；一道这应该可以防止它在新的偏见的影响下重蹈覆辙。” 149但这道保护现在不受过去影响的“屏障”依赖于经验性的、因此可能被证伪的关于人类心灵本质的主张。如果人们并不像孔多塞所想的那样天生理性呢？如果他误以为自然理性只是更多的偏见——只是欧洲社会武断的价值观和假设——那又会怎样？

This antipathy toward culture had a flip side. Since the goal was to start anew, received practices and, in particular, language appeared as a threat to science. Condorcet hoped that his universal algebra could be understood even if all languages were lost; like a Svalbard Seed Vault for knowledge, it would thus provide an insurance against lapses into darkness and chaos like the one that, no doubt, he feared France was facing during the Reign of Terror.148 His approach to achieving this universal intelligibility is modeled on Lockean sensationalism, which, he writes, “has forever imposed a barrier between mankind and the errors of its infancy; a barrier that should save it from relapsing into its former errors under the influence of new prejudices.”149 But this “barrier” protecting the present from the past depended on empirical and thus potentially falsifiable claims about the nature of the human mind. What if people were not as naturally rational as Condorcet thought? What if he mistook for natural reason what was, in reality, only more prejudice—only the arbitrary values and assumptions of European society?

孔多塞似乎没有意识到十八世纪抵御此类反对意见的最英勇努力。1784 年，大约在孔多塞撰写《哲学概要》的十年前，伊曼努尔康德宣称他的时代为“启蒙时代”。150康德和孔多塞一样，希望将人们从传统和教条中解放出来；他将启蒙理解为独立思考而不是依赖他人指导的勇气。然而，康德并未出现在孔多塞的进步叙事中。这种疏忽可能源于孔多塞的偏好或阅读的局限性。然而，康德的批判哲学确实与洛克的轰动主义有所不同，其最终效果是破坏了永久推翻文化的可能性——值得注意的是，也破坏了这种推翻文化的必要性，就像孔多塞想要用他的通用代数来实现的那样。

The eighteenth century’s most heroic effort to fend off such objections seemed not to enter Condorcet’s perception. It was in 1784, around a decade before Condorcet wrote the Sketch, that Immanuel Kant had declared his time an “age of enlightenment.”150 Kant wanted, just like Condorcet, to free people from traditions and dogmas; he construed enlightenment as the courage to think for oneself rather than relying on the guidance of others. Yet Kant makes no appearance in Condorcet’s narrative of progress. This oversight may have stemmed from Condorcet’s preferences or the limits of his reading. Yet Kant’s critical philosophy does point in another direction from Lockean sensationalism, and its ultimate effect was to undermine the possibility of—and, it is important to note, the necessity of—permanently overthrowing culture in the way Condorcet wanted to do with his universal algebra.

康德的著作在思想史上长期以来占据着模棱两可的地位。尽管他自认为是启蒙运动的支持者，但他也影响了那些通常被视为否定启蒙思想的浪漫主义思想家。这种模棱两可性导致克利福德·西斯金宣称，启蒙运动必须终结才能成功改变世界。151在《纯粹理性批判》 （ 1781）中，康德试图表明，所有理性存在都必须以十二个普遍范畴来思考，这些范畴决定了经验知识可能的条件。虽然这些范畴包括与几何相关的空间概念，但它们肯定不包括代数规则。然而，康德提供了一套新的术语，数学家可以用它们重新划定围绕符号方法的学科界限。康德的追随者借用他的思想来支持一种与他自己的明确观点相悖的立场——代数是“纯粹理性”的问题，源于思想本身，而不（与马塞雷、达朗贝尔和孔多塞相反）依赖于我们通过感官获得的知识。

Kant’s work has long occupied an ambiguous position in intellectual history. Self-consciously a proponent of Enlightenment as he was, he also influenced Romantic thinkers who are commonly seen as repudiating Enlightenment thought. This ambiguity has led Clifford Siskin to declare that the Enlightenment movement had to come to an end to succeed in changing the world.151 In Critique of Pure Reason (1781), Kant attempts to show that all rational beings must think in terms of twelve universal categories that determine the conditions in which empirical knowledge is possible. Although these categories include conceptions of space relevant to geometry, they certainly do not include the rules of algebra. Yet Kant provided a new set of terms by which mathematicians could redraw the disciplinary boundary around symbolic methods. Kant’s followers appropriated his ideas to support a position that went against his own explicit views—the idea that algebra is a matter of “pure reason,” stemming from thought alone and not (contra Maseres, d’Alembert, and Condorcet) having any dependence on the knowledge we receive through the senses.

虽然康德的批判主要在于界定纯粹理性所能达到的界限，但它也提供了一种巩固思考主体自主性的方法。与当时许多其他哲学家一样，康德坚持认为，哲学问题最好由自己去思考；如果学生不经过学习就死记硬背哲学家的体系，康德写道，推理“他已经掌握并保留了下来，也就是说，他学得很好，已经成为一个活生生的人的石膏模型。” 152在对死记硬背的怀疑上，康德与洛克并无太大区别，洛克写道，死记硬背结论的人“只会把他的理解变成别人木材的仓库”。153然而，这种个人自主的理想在经验领域面临着实际困难。例如，皇家学会倡导的实验科学依赖于信任：没有人能够亲眼目睹每一个实验，所以人们不得不相信别人的话。这种社会传播的知识形式与启蒙运动的信条“敢于求知”格格不入，因为它们暗示权威在个人学习过程中起着必要的作用。

Although Kant’s critique is largely about determining the bounds of what pure reason can do, it also provided a way of shoring up the autonomy of the thinking subject. Like many other philosophers of the period, Kant insisted that it was best to think through philosophical matters oneself; if a student memorizes a philosopher’s system without going through the reasoning, Kant writes, “he has grasped and retained, that is, he has learnt well and has become a plaster cast of a living person.”152 In this suspicion of rote learning, Kant is not too different from Locke, who wrote that a person who memorizes conclusions “makes his understanding only the warehouse of other men’s lumber.”153 Yet this ideal of individual autonomy faced practical difficulties in empirical fields. The experimental science promoted by the Royal Society, for instance, depended on trust: no one could witness every experiment firsthand, and so people had to credit the word of others. Such socially transmitted forms of knowledge sat uneasily with the Enlightenment credo “dare to know” because they implied that authority played a necessary role in how the individual learns.

康德实际上是通过将自主性限制在某些类型的知识上来应对这一困难的。在本书的第二部分，康德区分了他所谓的“理性”知识和“历史”知识。他写道，“如果拥有知识的人所知道的知识仅限于外界赋予他的部分，无论是通过直接经验、叙述还是（一般知识的）指导”；另一方面，如果知识完全源于理性，那么它就是理性的。” 154在语言、动物学、人类历史等真正具有历史意义的学科上，人们常常别无选择，只能依赖从他人那里获得的信息。对康德来说，理性知识有着根本的不同；它与感官经验的具体内容无关，因此，在理性领域中，人们有可能获得对他人的独立性，而不必抛弃自己所知道的一切。

Kant responded to this difficulty, in effect, by restricting autonomy to certain types of knowledge. In the second part of the book, Kant distinguishes what he calls “rational” and “historical” knowledge. Knowledge is “historical,” he writes, “if he who possesses it knows only so much of it as has been given to him from outside, whether through immediate experience or through narration, or also through instruction (of general knowledge)”; knowledge is rational, on the other hand, if it stems solely from reason.154 In subjects that are properly historical—language, zoology, human history—one often has no choice but to rely on information received from other people. Rational knowledge is fundamentally different for Kant; it has nothing to do with the specific content of sensory experience, and thus it is possible to obtain independence from others in the realm of reason without throwing out everything one knows.

为了支持这种设定理性界限的方式，康德提出了一对如今已为人熟知的术语：客观和主观。在康德之前，这些术语是经院逻辑词汇的一部分，在经院逻辑中，它们的含义与现在大致相反。正如牛津英语词典所说，客观的意思是“作为思想或意识的对象而存在，而不是真实存在”，而主观的意思是“与主体相关，因为主体中存在属性或特性；固有的；与事物的本质或现实相关；真实的，本质的。” 155康德之后，它们的含义发生了逆转：客观指的是所有人共有的知识方面，而主观指的是个人特有的知识方面。这些术语的使用存在一些歧义——主观也可以表示心灵内部，这并不明显排除确定性或普遍性——但它们提供了一种将学科知识与教育实践区分开来的有效方法。156学生对一个哲学体系的理解可能有所不同，程度也有所不同，但这种变化仅仅是“主观”问题，并不影响该体系本身的客观合理性。

In support of this way of setting the bounds of reason, Kant enlisted a now familiar pair of terms: objective and subjective. Prior to Kant, these terms had been part of the vocabulary of Scholastic logic, in which they had roughly the opposite of their current meanings. As the Oxford English Dictionary puts it, objective meant “Existing as an object of thought or consciousness as opposed to having a real existence,” whereas subjective meant “Relating to the subject as that in which properties or attributes inhere; inherent; relating to the essence or reality of a thing; real, essential.”155 After Kant, their meanings reversed: objective came to refer to the aspects of knowledge that were common to all people, whereas subjective meant those specific to the individual. There is some ambiguity in how these terms came to be used—subjective could also mean interior to the mind, which does not self-evidently exclude certainty or universality—but they provided a powerful way of delineating disciplinary knowledge from the practicalities of education.156 Students may understand a philosophical system in different ways and to different degrees, but this variation was a mere “subjective” matter that did not affect the objective rationality of the system itself.

虽然康德的纯粹理性领域非常有限，但他的追随者却不失时机地为其补充了广泛的逻辑、数学和诗意实践。康德思想启发逻辑学家将有效性标准与人类思维的经验事实区分开来。157代数中也发生了类似的划分，代数（至少在其理论形式上）被视为脱离经验现实的抽象科学。在十八世纪最后几年，这种重新构想导致了普遍代数思想的再次复兴，这一次的形式与孔多塞的方案截然不同。在 1798-99 年写成的一系列无标题片段中，德国诗人、采矿工程师和哲学家诺瓦利斯（生名弗里德里希·冯·哈登贝格）在康德耕耘的土地上复兴了莱布尼茨的普遍特征思想。21 世纪的译者大卫·W·伍德将其命名为《浪漫主义百科全书注释》。158诺瓦利斯的基本思想是将代数推广，使其不仅应用于量，也应用于质。159他的雄心壮志是（正如他在此类项目中一贯的）百科全书式的：他考虑将代数方法应用于音乐、形而上学和诗歌等各种领域。他宣称：“所有科学都应该成为数学。” 160

While Kant’s realm of pure reason is highly limited, his followers wasted no time in outfitting it with a wide range of logical, mathematical, and poetic practices. Kantian thought inspired logicians to divide standards of validity from empirical facts about human thought.157 A similar division took place in algebra, which came (at least in its theoretical forms) to be seen as an abstract science detached from empirical reality. In the final years of the eighteenth century, this refiguring led to another revival of the idea of a universal algebra, this time in a very different guise from Condorcet’s scheme. In the untitled series of fragments written in 1798–99 that their twenty-first-century translator David W. Wood has dubbed Notes for a Romantic Encyclopaedia, the German poet, mining engineer, and philosopher Novalis (born Friedrich von Hardenberg) revived Leibniz’s idea of a universal characteristic on the ground Kant had tilled.158 Novalis’s fundamental idea is to generalize algebra so as to apply to qualities as well as quantities.159 His ambitions were (as they always have been in such projects) encyclopedic: he considered applying algebraic methods to fields as diverse as music, metaphysics, and poetry. “All sciences,” he declares, “should become mathematics.”160

诺瓦利斯在笔记中赞许地引用了孔多塞的《素描》，但从风格上看，两者仿佛来自不同的世界。161如果说孔多塞煞费苦心地通过清晰的定义将代数植根于意义之中，那么诺瓦利斯则将其视为纯粹的符号之舞。在他1798年那篇名为《独白》的非凡文章中，他指出，即使是语言本身也无需指涉世界才能有意义：

Novalis quotes approvingly from Condorcet’s Sketch in his notes, but stylistically the two might as well have come from different worlds.161 If Condorcet took pains to ground algebra in the senses through clear definitions, Novalis viewed it as a pure dance of symbols. In the remarkable 1798 text titled “Monologue,” he suggests that even language itself needs no reference to the world to be meaningful:

但愿人们能明白，语言和数学公式是一样的。它们构成了一个独立的世界。它们只与自身互动，只表达自身奇妙的本质，正因如此，它们才如此富有表现力——正因如此，事物之间奇特的关系才在它们身上得以体现。162

If one could only make people understand that it is the same with language as with mathematical formulae. These constitute a world of their own. They play only with themselves, express nothing but their own marvelous nature, and just for this reason they are so expressive—just for this reason the strange play of relations between things is mirrored in them.162

这样的论述使得哲学家保罗·雷丁将诺瓦利斯置于莱布尼茨的原始计算主义传统之中。163然而，诺瓦利斯远非教条主义的莱布尼茨主义者。他追随康德的脚步，在数学知识和经验知识之间划了一条清晰的界线；从这个角度来看，莱布尼茨将两者杂糅在一起，缺乏条理。诺瓦利斯的普适代数学非但没有产生关于感官世界的知识，反而将看似经验的科学变成了纯粹的抽象事物。

Such statements have led the philosopher Paul Redding to place Novalis in the protocomputationalist tradition of Leibniz.163 Yet Novalis was far from a doctrinaire Leibnizian. Following Kant’s lead, he drew a hard line between mathematical and empirical knowledge; from this perspective, Leibniz was mixing the two together incoherently. Rather than produce knowledge about the world of the senses, Novalis’s universal algebra would turn seemingly empirical sciences into matters of pure abstraction.

从严格的康德主义视角来看，将纯粹理性与代数相提并论并不完全正确。就像约翰·戈特利布等其他唯心主义者一样与费希特不同，诺瓦利斯违背了康德关于纯粹理性界限的规定。结果，苏格兰逻辑学家威廉·汉密尔顿后来指责唯心主义者是披着康德外衣的莱布尼茨主义者。164无论是否合法，诺瓦利斯都在大胆地探索类似算法思维的东西。他在 1799-1800 年的一个片段中写道：“纯数学与数量无关。它是对名称的单纯研究——在有序概念操作的关系中变得机械化。它只能是任意教条的工具。” 165 “工具性”这个词在这里与其说是对工具主义的拥抱，不如说是与器乐的比较。就像没有歌词的音乐一样，代数并不涉及任何特定的东西，而是从其元素相关联的“运算”中获得意义。如果说孔多塞将代数的符号“象形文字”视为现在与过去之间的屏障，那么诺瓦利斯则借鉴康德，将这道屏障重新定位为存在于当下的理性知识与历史知识之间的屏障。代数是纯粹的思维，不依赖于任何概念内容，因此它具有应用于任何事物的潜力。

From a strictly Kantian perspective, this conflation of pure reason with algebra was not quite right. Like other idealists such as Johann Gottlieb Fichte, Novalis disobeyed Kant’s stipulations about the limits of pure reason. As a result, the Scottish logician William Hamilton would later accuse the idealists of being Leibnizians in Kantian clothing.164 Legitimately or not, Novalis was venturing boldly toward something like algorithmic thinking. “Pure mathematics,” he wrote in a 1799–1800 fragment, “has nothing to do with quantity. It is the bare study of names—become mechanical, in relationships of orderly conceptual operations. It must merely be arbitrarily–dogmatically instrumental.”165 The word instrumental here indicates less an embrace of instrumentalism than a comparison with instrumental music. Like music with no words, algebra is not about anything in particular, gaining its meaning instead from the “operations” by which its elements are related. If Condorcet saw the symbolic “hieroglyphs” of algebra as a barrier between present and past, Novalis, drawing on Kant, repositioned this barrier to lie between rational and historical knowledge as they exist here and now. Algebra is pure thought, having no dependence on any conceptual content, and it is thus potentially applicable to anything.

诺瓦利斯是一位非同寻常的人物，他关于代数的言论常常引发人们的困惑。但他的言论与该领域的发展方向并不冲突。正如诺瓦利斯宣称代数是符号的纯粹舞蹈一样，19世纪的代数学家们重新定义了他们的科学，使其只处理符号方程式中表达的运算，而不涉及数量概念。这一思想源于拉格朗日对代数普遍性的坚持，尽管后来的数学家发现拉格朗日的某些工作缺乏严谨性。在他1797年出版的《解析函数理论》（诺瓦利斯在其笔记中引用了该理论）中，拉格朗日尝试将微积分建立在他所谓的“函数算法”之上——这种用法很可能促成了“算法”一词在19世纪的广泛采用。166拉格朗日的“算法”基于一种称为泰勒级数的近似方法，该方法以 18 世纪早期数学家布鲁克·泰勒 (Brook Taylor) 的名字命名。给定一个函数f ，其值a无限可微，我们通常可以将该函数近似为f ( x ) ≈ f ( a ) + ( f ′( a )/1!)( x – a ) + ( f ′′( a )/2!)( x – a ) ² + ( f ′′′( a )/3!)( x – a ) ³ + · · ·，其中f ′ 是函数的一阶导数，f ″ 是二阶导数，依此类推。拉格朗日建议，我们不必使用莱布尼茨的无穷小dx，而是可以直接将导数定义为泰勒级数中出现的函数。

Novalis was an unusual figure, and his pronouncements about algebra were largely met with puzzlement. But his remarks were not out of line with the direction the field was taking. As Novalis declared algebra a pure dance of symbols, nineteenth-century algebraists would redefine their science to deal only with the operations expressed in symbolic equations, not with notions of quantity. This idea had roots in Lagrange’s insistence on the generality of algebra, although later mathematicians found some aspects of Lagrange’s work lacking in rigor. In his 1797 Theory of Analytical Functions (which Novalis cites in his notes), Lagrange presents an attempt to ground the calculus on what he calls “the algorithm of functions”—a usage that likely contributed to the wider adoption of the term algorithm in the nineteenth century.166 Lagrange’s “algorithm” is based on an approximation method called the Taylor series, named after the early eighteenth-century mathematician Brook Taylor. Given a function f that is infinitely differentiable for value a, we can often approximate the function as f (x) ≈ f (a) + (f ′(a)/1!)(x – a) + (f ′′(a)/2!)(x – a)² + (f ′′′(a)/3!)(x – a)³ + · · ·, where f′ is the first derivative of the function, f″ the second, and so forth. Instead of working with Leibniz’s infinitely small dx, Lagrange suggests, one could simply define the derivatives as the functions that appear in the Taylor series.

拉格朗日的“函数算法”拥有众多崇拜者，其中包括查尔斯·巴贝奇，他在剑桥推广拉格朗日思想，并将拉格朗日的概括方法扩展到博弈分析等其他领域。167然而，大约在1820年，奥古斯丁-路易·柯西和伯纳德·博尔扎诺破坏了拉格朗日的基础程序：可以构造反例，使得泰勒级数不收敛于原函数，这后来（公平与否）被视为拉格朗日理论的一个致命缺陷。168这一判断源于与导致阿贝尔拒绝欧拉关于发散级数的工作相同的认识论转变。虽然数学证明的标准在 19 世纪的大部分时间里仍在不断变化，但人们普遍不再接受达朗贝尔的“形而上学”方法。到 19 世纪后期，数学家们已经不再为如何定义诸如负数和无穷小等棘手的术语而烦恼；概念解释被视为次于形式定义和规则。169因此，孔多塞让符号本身可以理解，而无需使用任何语言的计划失去了紧迫性。现在，对于口头解释的持续需求可以被视为仅仅是一个“主观”问题，主要与教学法、说服力和数学社区的培养有关，但与代数本身无关。

Lagrange’s “algorithm of functions” had numerous admirers, among them Charles Babbage, who promoted Lagrangian ideas at Cambridge and extended Lagrange’s method of generalization to other areas such as the analysis of games.167 Around 1820, however, Augustin-Louis Cauchy and Bernard Bolzano would undermine Lagrange’s foundational program: it is possible to construct counterexamples in which the Taylor series does not converge to the original function, which came (fairly or not) to be seen as a fatal flaw in Lagrange’s theory.168 This judgment resulted from much the same epistemological shift that led Abel to reject Euler’s work on divergent series. Although standards of mathematical proof remained in flux through much of the nineteenth century, the turn away from d’Alembert’s “metaphysical” approach was general. By the late 1800s, mathematicians had stopped fretting over how to define such tricky terms as negative and infinitesimal; conceptual explanations came to be seen as secondary to formal definitions and rules.169 Condorcet’s project of making symbols comprehensible by themselves, without any use of language, thus lost its urgency. The continued need for verbal explanations could now be filed away as a merely “subjective” issue relevant mainly to pedagogy, persuasion, and the cultivation of a mathematical community, but not internal to algebra itself.

当然，结果并非所有数学家都成了康德主义者。数学家对康德的评价不一，许多人拒绝他的思想，甚至完全忽视他；即使是那些钦佩他的人，也更多地借鉴了他哲学的总体精神，而非他对数学的具体见解。170与其说是哲学上的转变，不如说是优先事项的改变。部分原因是人们认为法国大革命过于激进，1790 年代的改革让位于对平衡的呼声；科学不再被视为一种通过取代既定思维模式来使人们更加理性的手段，而是被视为一种方法或知识体系，必须与文学研究和情感培养保持平衡。启蒙运动改革的一些成果得以延续，最显著的是公制；度量衡在 19 世纪才进一步走向标准化。然而，与新兴的主观性范畴相关的问题开始得到新的温和处理。科学的进步不再意味着彻底清除头脑中旧的思维方式，就像孔多塞的通用代数所要做的那样；天才必须发挥其神秘的魔力，为思想提供呼吸的空间。

The outcome, to be sure, was not that all mathematicians became Kantians. Mathematicians had varying evaluations of Kant, with many rejecting his ideas or ignoring him altogether; even those who admired him drew more on the general spirit of his philosophy than on his specific opinions about mathematics.170 What happened was less a philosophical shift than a change in priorities. Due in part to the perceived excesses of the French Revolution, the reforms of the 1790s gave way to calls for balance; science came to be seen not as a means of making people more rational by replacing received modes of thinking but rather as a method or body of knowledge that must be balanced with the study of literature and the cultivation of sentiment. Some fruits of Enlightenment reform lasted, most notably the metric system; weights and measures would only move further toward standardization in the nineteenth century. Matters affiliated with the emerging category of subjectivity, however, came to be treated with a new softness. Advancing science no longer meant expunging old ways of thinking from the mind altogether, as Condorcet’s universal algebra was supposed to do; genius had to work its mysterious magic, thought given room to breathe.

如果说孔多塞的预言在某个方面成真了，那一定是在经济领域。在《纲要》中，他设想将形成“一个伟大的民族，其语言举世皆知，其商业关系遍布全球”。171他关于商业的宣言预示着国际贸易的兴起和全球货币等价体系的建立。然而，全球化并没有建立他所期望的兄弟情谊。事后看来，孔多塞的政治著作清楚地揭示了启蒙运动普遍主义的根本矛盾。孔多塞反对奴隶制（尽管这他不仅是一位种族主义者（这当然不意味着他没有种族主义），而且他支持性别平等。172这些承诺如何与尊重其他文化相协调——孔多塞政治纲领的倡导者如何能在不强迫所有人接受自己观点的情况下，在世界范围内促进平等？孔多塞完全站在这个问题的一边，信奉一种假定特定价值观优越性的平等形式。他谴责对非西方民族的殖民压迫，这种压迫源于“对其他肤色或信仰的人的残忍蔑视”，但他设想未来，“我们不再总是表现得像暴君或腐败者，而要成为他们有用的工具或慷慨的解放者。” 173

If Condorcet’s predictions came true anywhere, it was in the sphere of economics. In the Sketch, he envisions the formation of “a great people whose language is universally known and whose commercial relations embrace the whole area of the globe.”171 His proclamations about commerce herald the rise of international trade and the establishment of a worldwide system of monetary equivalences. Yet globalization did not establish the brotherhood he desired. With the benefit of hindsight, Condorcet’s political writings plainly expose the fundamental contradiction of Enlightenment universalism. Condorcet was an opponent of slavery (although this certainly does not mean he was devoid of racism), and he supported gender equality.172 How can these commitments be reconciled with a respect for other cultures—how can advocates of Condorcet’s political program foster equality worldwide without forcing their ideas on everyone? Condorcet comes down wholly on one side of this question, embracing a form of equality that presumes the superiority of a particular set of values. He condemns colonial oppression of non-Western peoples, which stemmed from a “murderous contempt for men of another colour or creed,” but he envisions a future time when, “no longer presenting ourselves as always either tyrants or corrupters, we shall become for them useful instruments or generous liberators.”173

说这种立场具有令人厌恶的含义，也不过分。征服者们一次又一次地自欺欺人地认为他们会被视为解放者。但认为仅仅通过将数学与政治分离就能避免孔多塞落入的陷阱，未免天真。传播自由的愿望与尊重现有文化的需要之间的冲突在18世纪曾被公开辩论，符号方法的倡导者就他们的体系在这些问题上如何失利表明了明确的立场。当数学失去了与经验的联系后，人们便可以宣称一个体系具有普遍性，而不必面对普遍主义所带来的严峻选择。这种非政治化的途径在布尔身上可见一斑，他的意图完全是静默主义的——并非要改变文化，而是要在既有文化之间找到一种“和谐”。布尔逻辑可以说是通用代数运动最伟大的实践成功，并且（尽管人们可能对布尔本人的功劳有多少争论不休）它确实改变了世界；但这种成功的代价是放弃了所有希望，即仅仅通过符号就能创造孔多塞在恐怖统治期间躲避雅各宾派时所梦想的新开端。

To say that this position has unsavory implications would be an understatement. Conquerors have time and again deluded themselves that they would be greeted as liberators. But it would be naïve to suppose that one can avoid the trap into which Condorcet fell simply by detaching mathematics from politics. The conflict between the desire to spread liberty and the need to respect existing cultures was debated openly in the eighteenth century, and advocates of symbolic methods took explicit stances as to how their systems fell out in regard to these issues. After mathematics lost its link to the empirical, it became possible to declare a system universal without facing up to the stark choices universalism entails. Such an apolitical approach may be discerned in Boole, whose intentions were altogether quietistic—not to change culture but to find a “harmony” between cultures as they already exist. Boolean logic is arguably the greatest practical success of the universal algebra movement, and (although one could dispute how much credit should go to Boole himself) it really did change the world; but this success came at the cost of abandoning all hope that mere symbols could create the new beginning of which Condorcet dreamed while in hiding from the Jacobins during the Terror.

休战

The Truce

在讨论十九世纪的代数时，数学史学家们曾写道“符号转向”。十八世纪，让·勒朗·达朗贝尔和孔多塞侯爵等数学家曾将符号视为一种用于分析世界的“特殊语言”；因此，符号方法的有效性归结于能否将符号与清晰的思想联系起来。到了1900年，人们开始从形式的角度来看待符号。数学的严谨性并非源于言语定义的清晰度，而是源于公理体系的构建，而公理体系的构建主要以内部一致性为判断标准；符号代数可以有多种解释，或者，甚至可以无需任何解释即可使用。1将这一发展与计算机联系起来并非不合时宜：站在符号转向前沿的是查尔斯·巴贝奇，他是法国数学思想的推动者，设计了从未完成的蒸汽动力计算机（称为分析机），以及工厂制度的直率辩护者。2

In discussing the algebra of the nineteenth century, historians of mathematics write of a “symbolic turn.” In the eighteenth century, mathematicians such as Jean Le Rond d’Alembert and the Marquis de Condorcet had thought of symbols as a “particular kind of language” for analyzing the world; the validity of symbolic methods thus came down to whether one could attach the symbols to clear ideas. By 1900, symbols had come to be viewed, instead, in terms of form. Mathematical rigor came not from the clarity of verbal definitions but from the construction of axiomatic systems judged primarily by their internal consistency; symbolic algebra could be open to multiple interpretations or, perhaps, used without any interpretation at all.1 To connect this development to computing machines is not anachronistic: at the forefront of the symbolic turn was Charles Babbage, promoter of French mathematical thought, designer of the never-completed steam-powered computers called analytical engines, and forthright apologist of the factory system.2

代数学的符号化转向与语言学思想中看似相反的运动相吻合。在上个世纪，像孔狄亚克神父这样的启蒙哲学家将所有语言，包括法语和代数学，都置于“自然理性”的规范之下；这样一来思维方式的改变引发了一场革命性的计划，即用更精确的语言取代现有的语言。在十九世纪，像威廉·冯·洪堡这样的语言学家则将语言视为一个民族“精神生活”的有机产物。3像塞缪尔·泰勒·柯勒律治这样的浪漫主义作家致力于瓦解弗朗西斯·培根在语言批判中隐含的词语与事物的对立；虽然柯勒律治认为个体作家有可能影响语言，但他认为这一过程是与传统的协商，而非从第一原则进行的重建。4关于谁创造了语言的争论被推到了边缘，因为语言学家们开始一致认为语言是一种集体创造，而不是像弗朗茨·博普所说的那样是“学者或僧侣”的发明。5语言非但没有阻碍科学进步，反而被视为一种有价值的东西，是一个国家文化的重要组成部分，必须加以保护和珍惜。6

The symbolic turn in algebra coincided with a seemingly contrary movement in linguistic thought. In the previous century, Enlightenment philosophers such as the Abbé de Condillac had held all languages, including both French and algebra, to the standard of “natural reason”; this way of thinking had given rise to the revolutionary program of replacing existing languages with something more precise. In the nineteenth century, philologists such as Wilhelm von Humboldt treated language, instead, as an organic product of the “mental life” of a people.3 Romantic writers such as Samuel Taylor Coleridge pushed to collapse the antithesis of words and things implicit in Francis Bacon’s critique of language; while Coleridge thought it was possible for individual writers to influence language, he saw this process as a negotiation with tradition rather than a reconstruction from first principles.4 The debates about who made language were pushed to the fringes, as philologists came to agree that language was a collective creation rather than an invention of “the learned or the priesthood,” as Franz Bopp put it.5 Far from a hindrance to scientific progress, language was now seen as something valuable, an important element of the culture of a nation that had to be preserved and cherished.6

本章论证了对本土语言的重新认识，在当时算法思维最重要的进步之一——布尔逻辑——中发挥了作用。乔治·布尔的符号逻辑系统开发于19世纪40年代，是搜索引擎以及计算机系统许多其他领域中使用的“与”和“或”运算符的前身；它使得基于代数（例如现代符号中的数学或代数）和逻辑的嵌套结构复合表达式的构建成为可能。布尔将自己的工作描述为“迈向哲学语言的一步”，呼应了G.W. 莱布尼茨、孔多塞以及他之前的许多人的主张。7但与他的前辈们的方案不同，布尔的逻辑系统并不承诺完全、自给自足地取代现有的语言；相反，它提供了抽象的符号形式，这些形式必须在其他学科的语言中进行解释才能变得有意义。我希望表明，这种方法反映了浪漫主义思想家（例如威廉·华兹华斯）所倡导的二元文化观，他们力求在科学方法的严谨性与唯有自然语言才能提供​​的诗意洞见和道德情感之间取得平衡。尽管这种二元论明显是反机械论的，但它最终赋予了符号操作的机械系统比启蒙运动时期笛卡尔和洛克的认识论所赋予的更大程度的独立于人类思维的自主性，从而使算法——这个词当时正接近其现代含义——在知识生产中占据了新的核心地位。

This chapter argues that this renewed appreciation for vernacular languages played a role in one of the period’s most important advances in algorithmic thinking: Boolean logic. Developed in the 1840s, George Boole’s system of symbolic logic is a precursor of the and and or operators used in search engines and in many other aspects of computer systems; it enabled the construction of compound expressions with a nested structure based on algebra, such as, in modern notation, (mathematics or algebra) and logic. By characterizing his work as “a step toward a philosophical language,” Boole echoed the claims of G. W. Leibniz, Condorcet, and many others before him.7 But unlike the schemes of his precursors, Boole’s logic system does not promise a complete, self-sufficient replacement for existing languages; instead, it provides abstract symbolic forms that must be interpreted within the languages of other disciplines to become meaningful. This approach, I hope to show, reflected a dualistic view of culture promoted by Romantic thinkers such as William Wordsworth, who sought to balance the rigor of scientific methods with the poetic insight and moral feeling that only natural language could provide. Although it was overtly anti-mechanistic, this dualism ultimately granted mechanical systems of symbol manipulation a greater degree of autonomy from human thought than the Cartesian and Lockean epistemologies of the Enlightenment had allowed them, thus enabling algorithms—a word that was then coming close to its modern sense—to take a newly central role in the production of knowledge.

布尔于1815年出生于林肯郡一个工人阶级家庭，几乎没有接受过正规教育。然而，他却是一位求知若渴的自学者，自学了现代和古典语言，以及代数和微积分，并最终成为一名受人尊敬的数学家；1849年，他被任命为他是爱尔兰科克皇后学院的首任数学教授，但从未获得过学位。在他的著作《逻辑的数学分析》（1847 年）和《思维规律的研究》（1854 年）以及 1848 年的一篇简短的期刊文章中，他提出了一种用代数符号表示逻辑命题的方法。8这个系统建立在19世纪代数发展的基础上，他在《数学分析》的开头总结道：“熟悉符号代数理论现状的人都知道，分析过程的有效性并不取决于对所用符号的解释，而仅仅取决于它们的组合规律。” 9从 19 世纪 10 年代开始，包括乔治·皮科克和布尔的导师邓肯·F·格雷戈里在内的新一代代数学家，根据对符号排列形式规则的遵循，重新定义了严谨性。10正如皮科克所说，代数符号“受定律支配，这些定律也必须支配并在一定程度上决定它们的解释，而不是相反。” 11这种逆转使得代数算子可以有多种解释，这些解释可能与数量有关，也可能无关。

Born in 1815 to a working-class Lincolnshire family, Boole had little formal education. He was, however, a voracious autodidact, teaching himself modern and classical languages as well as algebra and calculus, and he established himself as a respected mathematician; in 1849, he was appointed the inaugural professor of mathematics at Queens College in Cork, Ireland, without ever having earned a degree. In his books The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854), as well as in a brief 1848 journal article, he presents a method for representing logical propositions with algebraic symbols.8 This system built on nineteenth-century developments in algebra, which he sums up at the beginning of Mathematical Analysis: “They who are acquainted with the present state of the theory of Symbolical Algebra, are aware, that the validity of the process of analysis does not depend upon the interpretation of the symbols which are employed, but solely upon the laws of their combination.”9 Starting in the 1810s, a new generation of algebraists, including George Peacock and Boole’s mentor Duncan F. Gregory, redefined rigor based on the following of formal rules concerning the arrangements of symbols.10 Algebraic symbols, as Peacock puts it, are “governed by laws which must likewise govern, and to a certain extent determine, their interpretation, and not conversely.”11 This reversal opened the possibility that algebraic operators can have multiple interpretations that may or may not relate to quantity.

布尔的逻辑理论源于这样的洞见：这种新的灵活的代数形式可以应用于逻辑命题的分析。12在布尔的体系中，加法表示两个范畴的组合，因此例如h + z可能表示马或斑马这一类。与现代布尔逻辑的或运算符相反，布尔的加法运算符仅适用于不重叠的范畴；这是他坚持使他的体系与普通代数保持一致的必然结果。13乘法表示范畴的交集，因此br可能表示棕色兔子；减法表示排除子范畴，因此s − c可能表示不是眼镜蛇的蛇。该系统使用 1 表示论域中的一切，使用 0 表示无；因此 1 − s可能表示不是蛇的一切。布尔还提出了他的代数的第二种诠释，这种诠释处理真值而不是范畴；在这个解释中，1 代表始终为真，0 代表永不为真。14

Boole’s logical theory proceeds from the insight that this newly flexible form of algebra could be applied to the analysis of logical propositions.12 In Boole’s system, addition represents the combination of two categories, so that h + z, for instance, might mean the class of things that are either horses or zebras. In contrast to the or operator of modern Boolean logic, Boole’s addition operator only applies to categories that do not overlap; this is a necessary consequence of his insistence on keeping his system aligned with ordinary algebra.13 Multiplication represents the intersection of categories, so that br might represent brown rabbits; and subtraction represents the exclusion of a subcategory, so that s − c might represent snakes that are not cobras. This system uses 1 to represent everything in the universe of discourse and 0 to represent nothing; thus 1 − s might represent everything that is not a snake. Boole also presents a second interpretation of his algebra that deals with truth values rather than categories; in this interpretation, 1 represents always true and 0 never true.14

布尔体系的两种诠释都与数值代数非常相似，不同之处仅在于增加了一条定律：x (1 − x ) = 0，或者等价地，xx = x。布尔称之为对偶定律，它代表着一个公理，即任何范畴都不与其对立面重叠——没有任何东西可以既是蛇又不是蛇。添加这条定律会使逻辑方程的行为类似于线性微分方程，这使得布尔能够开发出一种通用的求解程序。为了改编他的一个例子，假设我们有这样的前提：“每个诗人都是一个天才”，我们需要知道这告诉了我们关于“天才的人”的什么信息。15我们将前提表示为p (1 − g ) = 0，这意味着，从更字面意义上解释，诗人 ( p ) 而不是天才的人 (1 − g )的实体类别为空；然后人们使用布尔方法求解g。16结果g = p + v ( 1 − p ) 表明“天才的人”类别包含所有诗人以及一定数量（用特殊符号v表示）的其他非诗人。

Both interpretations of Boole’s system maintain a close analogy with numerical algebra, differing only in the addition of one law: x(1 − x) = 0 or, equivalently, xx = x. This law, which he called the law of duality, represents the axiom that no category overlaps with its opposite—nothing can be both a snake and not a snake. Adding this law causes the logical equations to behave somewhat like linear differential equations, which enables Boole to develop a general procedure for solving them. To adapt one of his examples, suppose we are given the premise that “Every poet is a man of genius,” and we need to know what this tells us about “men of genius.”15 We express the premise as p(1 − g) = 0, which means, more literally interpreted, that the category of entities that are poets (p) and not men of genius (1 − g) is empty; one then solves for g using Boole’s method.16 The result, g = p + v(1 − p), states that the category “men of genius” contains all poets along with some indefinite number (represented by the special symbol v) of other beings who are not poets.

与莱布尼茨和孔多塞之前的方案非常相似，这个系统反映了一种愿望，即解决争端，使具有不同背景和信仰的人们达成一致。简而言之，布尔希望每个人都能和睦相处。这种对团结的渴望反映了他的个人背景——他是一个宗教上非正统的英国人，在以天主教为主的爱尔兰的一所新教大学任教，他总是渴望缓和紧张局势，避免卷入冲突。在《数学分析》的导言中，他表达了典型的早期维多利亚时代的信念，即“随着我们对所有真正科学认识的进步，在其各个分支之间将会发现一种日益增长的和谐。” 17与弗朗西斯·马塞雷斯和威廉·弗伦德一样，布尔同情一神论；他相信所有的宗教和哲学体系都表达了一个人类思想可以直接理解的单一真理。布尔的逻辑系统基于类似的思维，为（他认为）每种语言中已经存在的常见逻辑关系提供了符号表示。

Much like the previous schemes of Leibniz and Condorcet, this system reflected a desire to resolve disputes and bring people with disparate backgrounds and beliefs into accord. Boole, in short, wanted everyone to get along. This desire for unity reflected his personal background—a religiously unorthodox Englishman who taught at a Protestant university in predominantly Catholic Ireland, he was always eager to ease tensions and avoid becoming embroiled in conflict. In the introduction to Mathematical Analysis, he expresses the characteristically early Victorian “conviction, that with the advance of our knowledge of all true science, an ever-increasing harmony will be found to prevail among its separate branches.”17 Like Francis Maseres and William Frend, Boole had Unitarian sympathies; he believed that all religions and philosophical systems expressed a single truth that was directly accessible to all human minds. Boole’s logic system was based on similar thinking, offering a symbolic representation of the common logical relations that (he thought) already existed in every language.

很难否认这种思想与莱布尼茨的共鸣。布尔第一次发现莱布尼茨的逻辑著作是在1855年，当时莱布尼茨已经发表了他的体系。18他的妻子后来回忆说，当他发现这位莱比锡哲学家预见了他的二元性定律时，他表现出“孩子般的喜悦”。19与莱布尼茨一样，布尔也深深地沉浸于符号的力量之中。在《思维定律》中，他通过推测（尽管布尔补充了一些重要的警告）“外在事物的构成可能与内在思维的构成相对应”，呼应了莱布尼茨的和谐学说。20但布尔逻辑与莱布尼茨试图创建的逻辑有着根本的不同。布尔的体系不包含任何事物的百科全书目录；相反，它表达的形式关系不依赖于命题的经验内容。使其结果有意义是一个解释的问题，而母语仍然占据主导地位。十八世纪在语言政治上发生冲突的两种立场，即改革主义和传统主义，就这样达成了休战：人们可以对数学符号保持任意控制，同时仍然遵循用于解释数学符号的既定用法。

It is hard to deny the Leibnizian resonance of this thinking. Boole first found out about Leibniz’s logical work in 1855, after he had already published his system.18 His wife later reported that he reacted with “childlike delight” upon discovering that the Leipzig philosopher had anticipated his law of duality.19 Like Leibniz, Boole was deeply invested in the power of symbols, and in The Laws of Thought he echoes Leibniz’s doctrine of harmony by speculating (although Boole adds significant caveats) that “the constitution of things without may correspond to that of the mind within.”20 But Boolean logic is fundamentally different from what Leibniz was trying to create. Boole’s system would include no encyclopedic catalog of things; instead, it expressed formal relations that did not depend on the empirical content of propositions. Making its results meaningful was a matter of interpretation, in the undertaking of which the mother tongue still reigned. The two positions on the politics of language that clashed in the eighteenth century, reform and traditionalism, thus reached a truce: one could maintain arbitrary control over mathematical symbols while still deferring to received usage with regard to the words used to explain them.

本章介绍了进行这种重新配置的条件可能。我首先概述布尔时代的思想氛围，然后考察他的一些具体来源。伊曼纽尔·康德对布尔的影响重大；与同时期的其他一些数学家一样，布尔（误）解读了康德的纯粹理性思想，认为数学知识独立于物质世界而存在。21布尔的工作也反映了当时一个相对较新的价值范畴的兴起：文化。维多利亚时代早期的科学不再是用更理性的思维方式取代既定的思维方式，就像孔多塞和许多其他思想家在18世纪90年代激进运动期间试图做的那样。相反，当时的趋势是寻求一种平衡：科学方法必须与人们有机发展的语言和思维模式协同工作，才能产生既严谨又有意义的结果。我希望表明，十九世纪的这一思想转向是确立支撑现代算法理念的主客体分野的关键时刻。我首先从一个能说明这种转变的现象开始：心算。

This chapter offers an account of the conditions that made this reconfiguration possible. I begin with a general illustration of the intellectual climate of Boole’s time and then examine some of his specific sources. A major influence on Boole was Immanuel Kant; like some other mathematicians in the period, Boole (mis)interpreted Kant’s idea of pure reason to show that mathematical knowledge existed apart from the material world.21 Boole’s work also reflects the rise of what was then a relatively new value category: culture. Science, in the early Victorian period, was no longer about replacing established ways of thinking with more rational ones, as Condorcet and so many other thinkers had attempted to do during the radical ferment of the 1790s. The trend, instead, was toward seeking a balance: scientific methods must work together with people’s organically developing languages and modes of thought to produce results that were both rigorous and meaningful. This nineteenth-century intellectual turn, I hope to show, was a crucial moment in establishing the subject–object divide that undergirds the modern idea of algorithm. I begin with a phenomenon that illustrates the shift: mental calculation.

天才与精于算计的男孩

The Genius and the Calculating Boy

1818年，伦敦威斯敏斯特学校的两名学生宣布举行一场计算竞赛。22其中一位参赛者是十三岁的泽拉·科尔伯恩 (Zerah Colburn)，他是佛蒙特州卡伯特一位农民的儿子。科尔伯恩至少在两个方面引人注目：他天生两只手都多一根手指；从五岁开始，他就发展出了惊人的乘法、开方和心算大数因式分解的能力（图 4.1）。科尔伯恩的挑战者是威廉·罗恩·汉密尔顿 (William Rowan Hamilton)，一个比他小一岁的爱尔兰男孩。汉密尔顿后来成为十九世纪最重要的数学家之一。但科尔伯恩最终获胜。科尔伯恩几乎完全凭借自己掌握的技能，可以在几秒钟内求出 268,336,125 等大数的立方根，而且无需纸张。23

In 1818, two students at the Westminster School in London announced a calculating contest.22 One competitor was the thirteen-year-old Zerah Colburn, the son of a farmer from Cabot, Vermont. Colburn was remarkable in at least two ways: he was born with an extra finger on each hand, and, starting at the age of five, he had developed an astounding ability to multiply, extract the roots of, and factor large numbers in his head (figure 4.1). Colburn’s challenger was William Rowan Hamilton, an Irish boy about a year younger. Hamilton would go on to become one of the most important mathematicians of the nineteenth century. But Colburn was the victor. Using skills he developed almost entirely on his own, Colburn could determine the cube roots of numbers as large as 268,336,125 in a matter of seconds, with no need for paper.23

公众对这些精于计算的天才的反应，反映了人们对计算的普遍态度的转变。正如洛林·达斯顿所指出的，18 世纪的人们将计算天才视为智力的典范。24例如，1796 年的一篇杂志文章赞扬了神童杰迪戴亚·巴克斯顿，他通过数出《理查三世》演出中每个演员说的台词数等壮举，展现了“惊人的意志力” ；作者遗憾地表示，巴克斯顿没有雄心壮志，将他的才能用于伟大的目的。25另一位 18 世纪的神童是托马斯·富勒，他出生于西非，在弗吉尼亚的一个种植园里为奴。正如他向本杰明·拉什所展示的，富勒可以心算出时间间隔在任何两个时间点之间都可以精确到秒，甚至考虑闰年。26雅克·皮埃尔·布里索是与孔多塞等人共同创立的法国废奴协会的负责人，他认为这个例子证明非洲人和他们的奴隶主一样聪明，如果他们被允许享有自由和教育，他们将能够在所有科学领域表现出色。27这种对神童的推崇与启蒙运动的精神相一致，启蒙运动强调人具有理性思考的先天能力。在《新论文》中，莱布尼茨将数学神童的存在作为算术知识是天生的证明。28即使对于那些与约翰·洛克站在一起反对天赋知识的人来说，神童似乎也表明人类心灵具有可靠的自然推理能力，不需要外部帮助。

Public reactions to such calculating prodigies register shifts in general attitudes toward computation. As Lorraine Daston has shown, people in the eighteenth century viewed computational prodigies as the epitome of intelligence.24 A 1796 magazine article, for instance, praises the prodigy Jedediah Buxton for the “astonishing strength of mind” he showed through such feats as counting the number of words spoken by each actor in a performance of Richard III; the author regrets that Buxton did not have the ambition to apply his talents to great ends.25 Another eighteenth-century prodigy was Thomas Fuller, who was born in West Africa and enslaved in a Virginia plantation. As he demonstrated to Benjamin Rush, Fuller could mentally compute the interval between any two points in time down to the second, even accounting for leap years.26 Jacques-Pierre Brissot, head of a French abolitionist society cofounded with Condorcet and others, took this instance as a proof that Africans were just as intelligent as their enslavers and would be able to excel in all sciences if they were permitted liberty and education.27 This elevation of prodigies harmonized with the spirit of the Enlightenment, which emphasized the mind’s innate capacity for rational thought. In New Essays, Leibniz takes the existence of mathematical prodigies as proof that arithmetical knowledge is innate.28 Even for those who sided with John Locke against innate knowledge, prodigies seemed to indicate that the human mind had reliable natural reasoning abilities that did not require external aid.

泽拉·科尔伯恩最初也受到了同样的崇敬。弗朗西斯·贝利与这个小男孩相处了很长时间，他在1812年写道，科尔伯恩可以为数学做出巨大贡献，“他的头脑似乎天生就擅长阐明数学” 。29达特茅斯学院的一群“科学之友”提出了一项计划，让科尔伯恩接受通识教育，以便他能够充分发挥自己的技能。30他们不仅希望他长大后成为天才，还希望他已经发现了一些重要的东西：他可以快速分解大数因式分解，而这项任务当时还没有通用的方法。贝利在他1812年的文章中指出，“用指数表示量的幂和根”——即符号x ⁿ和ⁿ √ x −——的想法促成了通用“幂算法”的发展，从而推动了对数的发明。31贝利希望，“当他的思想更加成熟时”，科尔伯恩也许能够解释他是如何做到这一点的，从而为更多新“算法”的开发做出贡献。32

Zerah Colburn was initially treated with much the same reverence. Francis Baily, who spent a great deal of time with the boy, wrote in 1812 that he could make a great contribution to mathematics, “for the elucidation of which his mind appears to be peculiarly formed by nature.”29 A group of “friends of science” from Dartmouth College proposed a plan to give Colburn a liberal education so that he could put his skills to good use.30 The hope was not just that he would grow up a genius but also that he had already discovered something important: he could rapidly factor large numbers, a task for which no general method was known. In his 1812 article, Baily observes that the idea of “expressing the powers and roots of quantities by means of indices”—that is, the notations xⁿ and ⁿ√x−—led to the development of a general “algorithm of powers” that enabled the invention of logarithms.31 Baily hopes that, “when his mind is more cultivated,” Colburn may be able to explain how he does it and thus contribute to the development of more new “algorithms.”32

但事与愿违。科尔伯恩最终与其说是一位数学家，不如说是一个杂耍演员，他的父亲带着他周游世界，看“计算男孩”的演出要收22美分。科尔伯恩后来从事演艺事业，最终回到佛蒙特州当一名语言教师。他最终开始怀疑自己能力的实际价值。在他1833年的回忆录中，他以第三人称写道：“如果他认为，全面介绍他非凡的天赋以及他进行计算的方法，能对数学界有任何帮助，他早就应该发表了。” 33他回忆说，小时候，他告诉一位波士顿妇女，他不能与他人分享他的天赋：“上帝把它放在我的脑袋里，而我无法把它放在你的脑袋里。” 34可悲的是，他的命运与其说是作为一位伟大的数学家被铭记，不如说是作为心理学上的一件奇闻。在 1884 年的一篇题为《数学失败》的文章中，科尔曼·E·毕晓普 (Coleman E. Bishop) 写道，科尔本“以‘自然的方式’掌握了数学” 。35这对毕晓普来说不是一件好事，他以这个男孩的例子作为算术扼杀一般能力的证据心智发展：“他用得越多，就越愚蠢。” 36这是一个不友好的结论——科尔伯恩似乎至少具有正常的智力——但它表明人们的态度正在背离霍布斯旧的推理与计算的等式。

It was not to be. Colburn ended up less a mathematician than a sideshow attraction, his father parading him around the world and charging twenty-two cents to see “the Calculating Boy.” Colburn later took to acting and finally returned to Vermont to become a language teacher. He eventually came to doubt the practical value of his abilities. In his 1833 memoir, he writes in the third person: “Were it his opinion that a full account of his remarkable gift, and the methods by which he effected his calculations, would be of any service to the mathematical world, he should have published it long ago.”33 As a boy, he recalls, he told a Boston woman that he could not share his gift with others: “God put it into my head, and I cannot put it into yours.”34 His fate, sadly, was to be remembered less as a great mathematician than as a psychological curiosity. In an 1884 article entitled “The Mathematical Failure,” Coleman E. Bishop wrote that Colburn “had mathematics in ‘the natural way.’”35 This was not a good thing for Bishop, who takes the boy’s example as proof that arithmetic stifles the general development of the mind: “the more he used it the stupider he grew.”36 This was an uncharitable conclusion—Colburn seems to have been of at least normal intelligence—but it indicates how attitudes were turning away from the old Hobbesian equation of reasoning with computation.

论及十九世纪数学家的应有形象，威廉·罗文·汉密尔顿比科尔伯恩更具典范性。尽管他无法战胜“计算男孩”，但汉密尔顿（不要将他与逻辑学家威廉·汉密尔顿混淆）本人却是个早熟的孩子。他最初对语言最感兴趣，年轻时曾考虑投身于诗歌创作。37他后来回忆说，正是1818年与科尔伯恩的比赛激发了他学习数学的兴趣。38 1820年，科尔伯恩的剧团访问都柏林时，两人再次相遇，讨论了这位美国人发明的因式分解方法。事实证明，这种方法需要知道所有乘积以给定两位数结尾的两位数；科尔伯恩成功地记住了一个820行的表格。39汉密尔顿认为，当其中一个因数低于200时，这种方法很有效，但在更困难的情况下，它需要猜测。40这种对方法的反思使汉密尔顿有别于科尔伯恩，科尔伯恩（他在回忆录中承认）在解决那些无法通过他认为显而易见的程序解决的问题时遇到了困难。41对汉密尔顿来说，数学是关于发现程序，而不是执行程序。

In regard to what a mathematician was supposed to be in the nineteenth century, William Rowan Hamilton is more exemplary than Colburn. Although he could not beat the “Calculating Boy,” Hamilton (who is not to be confused with the logician William Hamilton) was himself a precocious child. He was initially most interested in languages, and as a young man he considered devoting himself to poetry.37 It was his 1818 contest with Colburn, as he later recounted, that first inspired him to study mathematics.38 The two met again when Colburn’s acting troupe visited Dublin in 1820, and they discussed the method the American had invented for factoring numbers. The method, it turned out, involved knowing all the two digit numbers whose products end in a given two digits; Colburn had effectively memorized a table of 820 rows.39 As Hamilton determined, this method worked well when one of the factors was below two hundred, but in more difficult cases it required guesswork.40 This reflectivity about methods set Hamilton apart from Colburn, who (as he admitted in his memoir) had difficulty with problems that could not be solved through procedures he found obvious.41 For Hamilton, mathematics was about discovering procedures, not performing them.

汉密尔顿最为人铭记的是他开发了一个名为四元数的代数系统，他在 1844 年至 1850 年间发表的一系列文章中对此进行了描述。四元数是复数的推广，复数是有实部和虚部的数，例如 4 + 6 i，其中i表示 −1 的平方根。由于复数包含两个分量，因此可以将其视为二维空间中的点。四元数通过添加两个虚部将维数扩展为四维：Q = w + ix + jy + kz。汉密尔顿的关键见解涉及制定四元数代数的一致规则。四元数的加法过程显而易见，但四元数的乘法会产生九种不同的虚部组合，这些组合的值需要确定。汉密尔顿提出了以下规则：

Hamilton is best remembered for developing an algebraic system known as the quaternion, which he described in a series of articles published between 1844 and 1850. Quaternions are a generalization of complex numbers, which are numbers with real and imaginary components, as in 4 + 6i, where i indicates the square root of −1. Since they contain two components, complex numbers may be thought of as points in a two-dimensional space. A quaternion expands the number of dimensions to four by adding two more imaginary components: Q = w + ix + jy + kz. Hamilton’s key insight involved working out consistent rules for quaternion algebra. The procedure for adding quaternions is obvious, but multiplying them produces nine distinct combinations of the imaginaries, whose values need to be determined. Hamilton proposes the following rules:

汉密尔顿证明这些规则的方式表明，自一个世纪前欧拉和达朗贝尔曾对负的定义提出过质疑。他预料到的第一个反对意见是四元数违反了交换律：ij ≠ ji。42他认为这种牺牲是必要的，因为它使系统在其他方面更加方便，并且他的方程应该以它们是否“得出具有足够一致性和优雅性的结果”来评判。43问题不在于为规则找到概念上的依据，而在于以这样的方式设置规则，使系统具有尽可能多的理想属性。

The way Hamilton justifies these rules indicates just how much the field of algebra had changed since a century before, when Euler and d’Alembert had disputed the definition of negative. The first objection he anticipates is that quaternions break the commutative law: ij ≠ ji.42 He argues that this sacrifice is necessary because it makes the system more convenient in other ways, and that his equations should be judged by whether they “conduct to results of sufficient consistency and elegance.”43 The problem is not to find a conceptual justification for the rules, but to set up the rules in such a way that the system has as many desirable properties as possible.

到了19世纪40年代，这种新的代数规则方法已经相当成熟。关于负数和虚数的长达数个世纪的争论逐渐平息：清晰的文字定义似乎不再像确定符号的使用规则那么重要。这些规则的制定中，德国数学家马丁·欧姆迈出了重要一步。（欧姆是电阻的测量单位，以马丁的兄弟格奥尔格命名，格奥尔格是一位数学家，后来成为物理学家，专攻电流研究。）从1816年开始，马丁·欧姆试图消除一直困扰算术的悖论，并发展出第一套能够解释虚数的一致a ^{b规则。在他1842年出版的著作}《数学分析的精神及其与逻辑系统的关系》（次年出版英译本）中，欧姆用某种康德式的术语解释了他的思想。44他写道，他对围绕虚数的困惑的解决方案基于这样的认识：数学表达式“并不代表量级（数量），而是代表心理行为（用系统语言来说：‘符号化运算’），这些心理行为彼此之间存在一定的关系，并以‘方程式’的形式表达出来。” 45所有的数字，无论是实数还是虚数，“本身都只是形式”，并且“整个数学分析仅仅用于给定形式的变换。” 46

By the 1840s, this new approach to algebraic rules was already well established. The centuries-long debates about negative and imaginary numbers had sputtered out: the clarity of verbal definitions no longer seemed as important as determining the rules by which symbols were to be used. An important step in working out these rules is attributable to the German mathematician Martin Ohm. (The ohm, a unit of measurement for electrical resistance, is named after Martin’s brother Georg, a mathematician and later physicist who went on to specialize in the study of electric current.) Beginning in 1816, Martin Ohm attempted to eliminate the paradoxes that continued to plague arithmetic, developing the first set of consistent rules for a^b that account for imaginaries. In his 1842 book The Spirit of Mathematical Analysis and its Relation to a Logical System (published in an English translation the following year), Ohm explains his thinking in somewhat Kantian terms.44 His solution to the confusions that surrounded imaginary numbers is based, he writes, on the realization that mathematical expressions “do not represent magnitudes (quantities), but mental acts (in systematic language: ‘symbolized operations’), which stand in certain relations to one another, that are enunciated in ‘equations.’”45 All numbers, real or imaginary, are “nothing but forms per se,” and “the whole of mathematical analysis is solely employed in the transformation of given forms.”46

通过将计算重新定义为“运算”而非概念定义，欧姆打破了那种唯心主义的观点，即数学表达式必须指代源自我们对空间和时间的理解或直觉的思想。加法和减法变成了“纯粹的形式”，“只不过是构造a + b和a − b这些形式”，也就是说，“客观地考虑，仅仅是把它们写下来” 。47按照这种观点，数学推理的正确性只能通过检查一个人所做的物理标记来判断；诸如人们对这些符号含义的理解等“主观”考虑则是次要的。欧姆将他的形式主义版本视为对主流智慧的拒绝，他声称，在他于1822年首次发表《完美后果数学体系的尝试》一书时，其他几位数学家称他的想法“疯狂” 。4819 世纪 40 年代，符号方法逐渐战胜了从笛卡尔到孔多塞时期盛行的概念方法。

By reframing calculation in terms of “operations” rather than conceptual definitions, Ohm breaks with the sensationalist view that mathematical expressions must refer to ideas derived from our understanding of or intuitions about space and time. Addition and subtraction become matters of “mere form,” consisting of “nothing more than the construction of these forms a + b, and a − b” which is, “objectively considered, the mere writing of them down.”47 The correctness of a mathematical inference, in this view, can be determined solely through an examination of the physical marks a person makes; “subjective” considerations, such as what one thinks the symbols mean, are secondary. Ohm represents his version of formalism as a rejection of the prevailing wisdom, claiming that, upon the first publication of his Attempt at a Perfectly Consequential System of Mathematics in 1822, several other mathematicians declared his ideas “insane.”48 By the 1840s, symbolic methods were winning out over the conceptual ones that had reigned from the time of René Descartes to that of Condorcet.

在英伦三岛，这种对符号的接受尤为突然。剑桥分析学会由当时还是本科生的查尔斯·巴贝奇和爱德华·弗伦奇·布罗姆黑德（不是拼写错误）于 1811 年创立，旨在在英国推广符号方法。49该学会声称其宗旨是倡导使用莱布尼茨的微积分符号，由于英国人对艾萨克·牛顿的民族主义偏好，莱布尼茨符号（至少在他们看来）在英国被不公平地忽视了。50 1816年，巴贝奇（与学会的另外两名成员约翰·赫歇尔和乔治·皮科克一起）出版了法国数学家西尔维斯特-弗朗索瓦·拉克鲁瓦编写的微积分教科书的英译本，以推广大陆数学家的符号方法。分析学会对代数的语言方面投入了大量的注意力，尽管其成员在抱负方面比孔多塞更为慎重。在1821年的一次题为《论符号对数学推理的影响》的演讲中，巴贝奇大量引用了法国思想家约瑟夫·玛丽·德·热兰多（Joseph Marie de Gérando）的观点。我在第三章中提到过他对哲学语言体系的批评。对巴贝奇来说，代数符号与词语有着根本的不同，因为它们的定义很简单，没有任何无关的含义来分散对主题的注意力。

This embrace of symbols was especially abrupt in the British Isles. The Cambridge Analytical Society, founded in 1811 by the then undergraduates Charles Babbage and Edward Ffrench Bromhead (not a misspelling), promoted symbolic methods in Britain.49 The stated purpose of the society was to advocate the use of Leibniz’s notation for calculus, which had (at least in their view) been unfairly overlooked in England because of a nationalistic preference for Isaac Newton.50 In 1816, Babbage (along with two other members of the society, John Herschel and George Peacock) published an English translation of a calculus textbook by the French mathematician Sylvestre-François Lacroix as a way of promoting the symbolic approach of Continental mathematicians. The Analytical Society devoted a great deal of attention to the linguistic aspects of algebra, although its members were more measured in their ambitions than Condorcet. In an 1821 talk titled On the Influence of Signs in Mathematical Reasoning, Babbage extensively quotes from the French thinker Joseph Marie de Gérando, whose criticisms of philosophical language schemes I mentioned in chapter 3. For Babbage, algebraic symbols are fundamentally different from words because their definitions are simple, including no extraneous meanings to distract from the matters at hand.

剑桥分析学会迷恋（在某种程度上，这种迷恋在拿破仑战争期间具有政治危险性）18 世纪晚期的法国传统，即将代数视为一种“分析”。然而，皮科克后来在这一传统之上迈出了重要一步。在 1830 年出版的《代数学论述》中，皮科克指出，诸如 + 和 − 之类的符号不必受解释它们的词语的限制：“将加法和减法的名称强加于这些运算，甚至直接从一门对其含义和应用有充分理解和严格限制的科学中推导出来，都不会对一门科学的结果产生任何影响，因为该科学只根据确定的定律来研究符号和符号的组合，而这些定律与符号本身的具体值完全无关。” 51皮科克的理论部分是为了解决围绕负数的难题，他表明，尽管代数最初是在算术的基础上发展起来的，但它不必受这个起点的限制。

The Cambridge Analytical Society was enamored (in a way that was politically dangerous during the Napoleonic Wars) with the late eighteenth-century French tradition that viewed algebra as a form of “analysis.” Peacock, however, would later take a major step beyond this tradition. In his 1830 Treatise on Algebra, Peacock argues that symbols such as + and − need not be constrained by the words used to explain them: “The imposition of the names of Addition and Subtraction upon such operations, and even their immediate derivation from a science in which their meaning and application are perfectly understood and strictly limited, can exercise no influence upon the results of a science, which regards the combinations of signs and symbols only, according to determinate laws, which are altogether independent of the specific values of the symbols themselves.”51 Peacock’s theory is meant in part to resolve the difficulty surrounding negative numbers by showing that, even though algebra was originally developed on the basis of arithmetic, it need not be limited by this starting point.

这一转变与“算法”一词的首次使用（类似于其现代含义）相吻合。尽管该词长期以来一直用于指代微积分方法，但现在它逐渐成为使用符号执行的程序的统称。《牛津英语词典》 英语中该含义的首次出现可以追溯到 1811 年；一个不可否认的例子出现在 1836 年版的查尔斯·赫顿 (Charles Hutton) 的《数学教程》中，该书指导我们“应用霍纳算法”，意为变换多项式的过程。52该术语更引人注目的使用出现在波兰数学家兼哲学家约瑟夫·马利亚·霍内-弗龙斯基 (Józef Maria Hoëné-Wroński) 的著作中，他宣称“不定求和算法……可能只是通用算法体系的众多组成部分之一，这些通用算法以其集体的力量，可以涵盖所有可能的数量生成，从而构成科学的绝对体系。” 53正如弗龙斯基的一位追随者所定义的，算法是“表示为构造数值数量而执行的操作的一般形式”，例如a + b = c。54尽管弗龙斯基在某种程度上游离于数学主流之外，但这种思想却与当时的普遍趋势相符。与欧姆一样，弗龙斯基认为运算优先于数量概念；“算法”不仅被视为实用的计算方法，也被视为一种使无理数等新数字类型变得可及的途径。

This shift coincided with the first uses of algorithm in something like its modern sense. Although the word had long been used to refer to the method of differential calculus, it was now coming to serve as a general term for procedures performed using symbols. The Oxford English Dictionary dates the first instance of this sense in English to 1811; an undeniable instance appears in the 1836 version of Charles Hutton’s Course of Mathematics, in which we are instructed to “apply Horner’s algorithm,” meaning a procedure for transforming a polynomial.52 An even more striking use of the term appears in the work of the Polish mathematician–philosopher Józef Maria Hoëné-Wroński, who declared that the “algorithm of indefinite summation . . . may only be one of the divers constituent parts of a system of universal algorithms, which, in their collective strength, might embrace all the possible generations of quantities, and form thus the absolute system of the science.”53 As one of Wroński’s followers defined it, an algorithm is a “general form that indicates operations to execute for constructing a numeric quantity,” such as a + b = c.54 Although Wroński existed somewhat outside of the mathematical mainstream, this thinking was in line with a general trend. Like Ohm, Wroński gave operations primacy over conceptions of quantity; “algorithms” were coming to be seen not just as practical methods of computation but also as the means by which new types of number such as irrationals become accessible.

但值得注意的是，算法的含义在十九世纪仍然相当模糊。1845 年的大都会百科全书指出（引用 Michael Stifel 1540 年代的著作），算法一词有时用于“表示任何种类的符号，目的是表达数字或数量之间的指定关系”，例如分数的a / b符号。55请注意，此算法不是指过程：而是指符号。虽然有些作者确实将算法与符号区分开来，但这个词仍然与代数符号紧密相关。这种联系一直持续到本世纪末，例如 James Byrnie Shaw 1895 年出版的《数学：算法的科学》一书，该书使用这个词来指代类似代数的系统，例如四元数。56类似算法思维的东西在十九世纪中叶开始出现，但算法的概念还没有完全脱离代数。

It should, however, be noted that the meaning of algorithm was still rather fluid in the nineteenth century. The 1845 Encyclopædia Metropolitana states (citing Michael Stifel’s work from the 1540s) that the word algorithm is sometimes used “to denote any species of notation whatever for the purpose of expressing the assigned relations of numbers or quantities to each other,” such as the a/b notation for fractions.55 Note that this algorithm does not refer to a procedure: it refers to a notation. While some authors did distinguish algorithms from notations, the word was still strongly associated with algebraic symbolism. This association persisted through the end of the century, as in James Byrnie Shaw’s 1895 book Mathematics: The Science of Algorithms, which uses the word to refer to algebra-like systems such as quaternions.56 Something like algorithmic thinking was beginning to emerge in the mid-nineteenth century, but the idea of algorithm had not fully gained its independence from algebra.

新方法看似机械，但也引发了不少不适。虽然代数涉及“符号操作”如今已是老生常谈，但这个短语在19世纪却很新颖，常常被用来嘲讽现代代数的肤浅。例如，在伦敦皇家学会宣读的一篇论文中，詹姆斯·布斯牧师严厉批评英国数学教师注重“符号操作的灵活灵巧”，而不是“对原理的了解和对研究方法的熟悉”。57约翰·文恩在其1866年的著作《机会逻辑》中也以类似的负面方式使用了这个词，质疑皮埃尔-西蒙拉普拉斯的归纳理论可以提供比“符号操作公式”更有价值的东西。58这些评论表明了阶级的恐惧，即数学成为一种体力活动而不是智力活动——“操作”主要指身体行为，特别是用手的动作——以及对数学理论脱离现实的担忧。

There was also a fair amount of discomfort about the seemingly mechanical nature of the new methods. While it is now a cliché that algebra involves the “manipulation of symbols,” this phrase was novel in the nineteenth century, and it often served as a way of mocking the perceived shallowness of modern algebra. For instance, in a paper read at the Royal Society of London, the Reverend James Booth castigates English mathematics teachers for focusing on “nimble dexterity in the manipulation of symbols” rather than on “the knowledge of principles and familiarity with methods of investigation.”57 John Venn uses the phrase in a similarly negative way in his 1866 book Logic of Chance, questioning whether Pierre-Simon Laplace’s theory of induction can provide anything more valuable than “formulæ for the manipulation of symbols.”58 These comments suggest a class-based fear of mathematics becoming a physical activity rather than an intellectual one—“manipulation” referred primarily to physical acts, especially with the hands—as well as an apprehension that mathematical theories were becoming disconnected from reality.

避免陷入“纯粹机械”的境地是智力发展的问题，因此也是教学的问题：人们不能盲目遵循规则，而必须掌握规则的原理。因此，当巴贝奇和汉密尔顿之流在设计符号处理的机械方法时，那个时期的教科书通常延续了上个世纪对概念基础的强调。萨拉·里卡多·波特 1835 年出版的《算术对话》（后来修订为《理性算术》）根据洛克原则教授计算。这本书的目的不是解释“纯粹的数字机械运算”，而是教授“与理性培养相结合的科学”。59这本书的主体是一个小男孩和他母亲之间的对话，他的两个兄弟姐妹偶尔会插话；这种对话旨在引导学生直观地了解算法规则，而无需直接陈述它们。艾达·洛夫莱斯关于教学的信件也同样强调掌握程序背后的原理。1834年，她写信给学生安娜贝拉·艾奇逊，谈到如今被称为欧几里得算法的东西：“我特别想知道，你是否能满意地推导出最大的共同度量——我的意思是，这个过程的基本原理。你必须彻底理解这个运算的原理。” 60洛夫莱斯夫人是当时上流社会教育家的典型代表，她认为数学并非一种机械技能，而是一种精神“修养”，其主要目的是培养严谨的思维习惯。61

Avoiding this lapse into the “merely mechanical” was a matter of intellectual development and, hence, of pedagogy: rather than follow rules blindly, one had to grasp their rationales. Thus, while the Babbages and Hamiltons of the world were designing mechanical methods of symbol manipulation, the textbooks of the period generally continued the previous century’s emphasis on conceptual underpinnings. Sarah Ricardo Porter’s 1835 Conversations on Arithmetic, later revised under the title Rational Arithmetic, teaches computation on Lockean principles. Instead of explaining “the mere mechanical operations of numbers,” the book aims to teach “the science, in combination with the cultivation of the reason.”59 The body of the book takes the form of a conversation between a young boy and his mother, with occasional interjections by his two siblings; this dialogue is supposed to lead the student to intuit the rules of algorism without stating them directly. Ada Lovelace’s letters on pedagogy reveal a similar emphasis on grasping the principles behind procedures. In 1834, she wrote to her student Annabella Acheson about what is now called the Euclidean algorithm: “I wish particularly to know if you make out the greatest common measure—I mean the rationale of the process—to your satisfaction. It is important that you should thoroughly understand the principles of this operation.”60 Lady Lovelace, typically of upper-class educators in the period, treated mathematics not as a mechanical skill but as a form of mental “cultivation” whose primary purpose was to instill habits of rigorous thinking.61

正如洛夫莱斯的例子所示，十九世纪的文化观念与我们看来的算法思维并不矛盾。毕竟，洛夫莱斯本人在 1843 年发表了有时被称为第一个真正的算法的东西：一种计算伯努利数的程序，旨在作为巴贝奇从未完成的计算机器的程序。62尽管这些“操作”可能是机械的，但理解它们背后的想法需要以非机械的方式思考，为想象力和直觉留出空间。这确实是泽拉·科尔伯恩的结论。他写道，心灵一直是“一台拥有巨大力量的机器”，但仅仅让机器自行运转是不够的；真正的进步需要心灵“通过适当的文化来培养、提炼和调节”。63这种文化不能简化为机械规则；必须让它有机地生长。不必在意背景中分析引擎的轰鸣声。

As the example of Lovelace shows, the nineteenth-century idea of culture was not at all incompatible with what looks to us like algorithmic thinking. It was, after all, Lovelace herself who published (in 1843) what is sometimes characterized as the first true algorithm: a procedure for computing Bernoulli numbers intended as a program for Babbage’s never-completed computing engine.62 Mechanical as such “operations” may be, understanding the ideas behind them required thinking in a nonmechanistic way that made room for imagination and intuition. Such, indeed, was Zerah Colburn’s conclusion. The mind, he writes, has always been “a machine of gigantic powers,” yet it is not enough to let the machine run on its own; true progress requires the mind to be “raised, refined, and regulated by suitable culture.”63 This culture cannot be reduced to mechanical rules; it must be allowed to grow organically. Never mind the whir of the analytical engine in the background.

健康的分离

A Wholesome Separation

据报道，布尔曾说过：“真正的数学家一定不仅仅是一名数学家，他还必须具备诗人的素质。” 64布尔的确写了大量诗歌，诗歌在他代数化逻辑的基本原理中发挥了作用。正如丹尼尔·布朗在其对维多利亚时代科学家诗人的研究中表明的那样，数学与诗歌的并置在当时与其说是悖论，不如说是一种常见现象。65我已经提到过诺瓦利斯声称“代数是诗歌”，以及威廉·罗恩·汉密尔顿对成为诗人的渴望。众所周知，阿达·洛夫莱斯在给母亲的信中祈求“诗意的科学”，而埃德加·爱伦·坡的短篇小说《失窃的信》则赞扬了既是诗人又是数学家的人物所具有的卓越智慧。66在本世纪后期，卡尔·魏尔斯特拉斯——微积分严格形式化的核心人物——评论说：“一个不具备诗人气质的数学家永远不会成为一个完整的数学家。” 67特别是诗歌应该提供什么是“单纯的数学”无法提供的？

“A real mathematician,” Boole reportedly said, “must be something more than a mere mathematician, he must be also something of a poet.”64 Boole did, indeed, write a large number of poems, and poetry played a role in his rationale for algebraizing logic. As Daniel Brown has shown in his study of Victorian scientist-poets, the juxtaposition of mathematics and poetry was less a paradox than a commonplace at the time.65 I have already mentioned Novalis’s claim that “Algebra is poesy” and William Rowan Hamilton’s dalliance with becoming a poet. Ada Lovelace famously asked for “poetical science” in a letter to her mother, and Edgar Allan Poe’s short story “The Purloined Letter” extols the superior intellect of a character who is both a poet and a mathematician.66 Later in the century, Karl Weierstrass—a central figure in the rigorous formalization of calculus—remarked that “a mathematician who is not somewhat of a poet will never be a complete mathematician.”67 What was poetry, in particular, supposed to supply that “mere mathematics” could not?

就布尔而言，思考一下他当时可能读的是哪种诗歌会有所帮助。布尔的诗歌品味偏向形式主义；他最喜欢的诗人是但丁，并热情洋溢地赞扬约翰·弥尔顿。他认为奥古斯都王朝被浪漫主义者不公正地诽谤。然而，布尔自己的诗歌却具有一种对话式的特质，肖恩·露西曾将其与威廉·华兹华斯的后期作品相提并论。68华兹华斯的影响在 19 世纪40年代如此广泛，以至于很难想象布尔不熟悉他，而且在布尔的散文作品中可以清楚地看到华兹华斯思想的痕迹。华兹华斯与他有时的合作伙伴塞缪尔·泰勒·柯尔律治一起，致力于在科学的分析方法与这些方法所忽略的道德方向感之间取得平衡，而布尔在描述符号逻辑的局限性时正是采用了这种划分。诗歌表达了现实生活中支配人类行为的情感依恋，但形式逻辑却将其抽象出来；因此逻辑和诗歌表现为一个整体的两半。

In the case of Boole, it is helpful to consider what poetry he was likely reading. Boole’s tastes in poetry hewed toward the formal; his favorite poet was Dante, and he effusively praised John Milton. He thought that the Augustans were unfairly maligned by the Romantics. Boole’s own poetry, however, has a conversational quality that Seán Lucy has likened to the later work of William Wordsworth.68 Wordsworth’s influence was so widespread by the 1840s that it is hard to imagine that Boole was unfamiliar with him, and there are clear traces of Wordsworthian thinking in Boole’s prose writings. Wordsworth, along with his sometime collaborator Samuel Taylor Coleridge, pushed for a balance between the analytical methods of science and the sense of moral direction that those methods left out, and Boole employed just this division when he described the limits of symbolic logic. Poetry expressed the emotional attachments that governed human behavior in actual life but that formal logic abstracted out; logic and poetry thus appeared as two halves of a whole.

考察华兹华斯的科学观，有助于理解诗歌在十九世纪数学家中扮演的角色。通俗的文学史版本将华兹华斯与浪漫主义的观点相提并论，认为浪漫主义是对科学的敌视。他1798年的诗歌《逆转的局面》强化了这种解读，他在诗中写道：“我们的干预智力 / 扭曲了事物美丽的形式；/——我们为了解剖而杀人。” 69鉴于此种言论，浪漫主义运动常常被视为对科学导致的世界“祛魅”的一种反应。但十八世纪的科学与今天已不再相同。华兹华斯在该书的“广告”中表示，《逆转局势》是对一位过于沉迷于“现代道德哲学书籍”的朋友的斥责。70 这里指的是洛林·达斯顿所说的“道德科学”——一种对道德的解剖，其基础是过分相信分析方法可以使人有德。71华兹华斯的争论与其说是针对科学本身，不如说是针对科学对文化领域的这种侵入。

A consideration of Wordsworth’s views of science sheds light on the role poetry played for nineteenth-century mathematicians. The popular version of literary history has assimilated Wordsworth to the view of Romanticism as antagonistic toward science. This reading is reinforced by his 1798 poem “The Tables Turned,” in which he states that “Our meddling intellect / Mis-shapes the beauteous forms of things; / —We murder to dissect.”69 On account of such statements, the Romantic movement is often viewed as a reaction against the “disenchantment” of the world produced by science. But science did not mean the same thing in the eighteenth century that it means today. Wordsworth states in the “Advertisement” to the book that “The Tables Turned” was a rebuke to a friend who was too attached to “modern books of moral philosophy.”70 The reference is to what Lorraine Daston has called “the moral sciences”—a dissection of morality underwritten by an excessive confidence that analytical methods can make people virtuous.71 Wordsworth’s quarrel is less with science per se than with such incursions of science into the realm of culture.

与第一代英国浪漫主义者中的其他人一样，华兹华斯最初支持法国大革命，并于 1791 年前往法国，参与他所认为的世界变革。他后来讲述了革命如何被视为一个将激进理想付诸实践的机会，“不是在乌托邦，不是在地下田野，也不是在某个秘密岛屿，天知道在哪里！/而是在这个世界上，也就是我们所有人的世界——最终，在这个世界上，/我们要么找到幸福，要么什么也得不到！” 72但现实并没有达到这些期望。法国陷入暴力，英国改革者的计划令人失望。18 世纪 90 年代的功利主义者培养了一批第二代知识分子，他们的父母试图将他们塑造成完全理性的人，这给他们带来了精神创伤：威廉·戈德温和他的女儿玛丽·雪莱，詹姆斯·密尔和他的儿子约翰·斯图尔特·密尔。73海丝特·斯坦霍普夫人也说，她的父亲（她给他起了个绰号叫“逻辑学家”）专横地强迫她进行逻辑思考。74道德科学曾试图改善人类，但事实证明，人类比理论所揭示的要复杂得多，因此科学的社会功能需要重新思考。

Like others in the first generation of British Romantics, Wordsworth initially supported the French Revolution, and he traveled to France in 1791 to participate in what he saw as a transformation of the world. He later recounted how the Revolution seemed an opportunity to put radical ideals into effect, “Not in Utopia, subterraneous Fields, / Or some secreted Island, heaven knows where! / But in the very world, which is the world / Of all of us,—the place where in the end / We find our happiness, or not at all!”72 The reality fell short of these expectations. France descended into violence, and the projects of English reformers were disappointing. The utilitarians of the 1790s raised a cohort of second-generation intellectuals who were traumatized by their parents’ attempts to shape them into perfectly rational beings: William Godwin with his daughter Mary Shelley, James Mill with his son John Stuart Mill.73 Lady Hester Stanhope, too, reported that her father (whom she called by the nickname “The ‘Logician’”) was overbearing in attempting to force her to think logically.74 The moral sciences had sought to improve people, but people had proven more complex than the theories had let on, and so the social function of science needed rethinking.

华兹华斯关于科学本质的最明确的表述之一出现在 1800 年版《抒情歌谣集》的序言中，这部诗集是他与柯尔律治合著的。在脚注中，华兹华斯拒绝了“诗歌与散文的对立”，而主张“诗歌与科学的更具哲学性的对立”，正如他在修订版序言中所阐明的那样，这意味着处理感受的语言和处理事实的语言之间的区别。75对华兹华斯来说，最好的诗歌语言是与自然界“持久”的经验有着深刻联系的语言；因此，它比约翰·德莱顿和亚历山大·蒲柏的诗歌“更持久、更具哲学性的语言”，后者的诗歌受到“虚假的精致或任意的创新”的束缚。76华兹华斯的“任意创新”的问题在于，观点认为，其结果缺乏英语等自然语言的情感共鸣，而英语与人类社会的生活紧密相连，并承载着数百年的历史。

One of Wordsworth’s most explicit statements on the nature of science appears in the preface first added to the 1800 edition of Lyrical Ballads, the collection he coauthored with Coleridge. In a footnote, Wordsworth rejects the “contradistinction of Poetry and Prose” in favor of “the more philosophical one of Poetry and Science,” meaning, as he clarifies in the revised version of the preface, a distinction between language that deals with feeling and language that deals with facts.75 The best poetic language, for Wordsworth, is that which has a deep connection to the “durable” experiences of the natural world; it thus provides “a more permanent and a far more philosophical language” than that of John Dryden and Alexander Pope, whose poetry is hobbled by “false refinement or arbitrary innovation.”76 The problem with such “arbitrary innovation,” from Wordsworth’s perspective, is that its results lack the emotional resonances of natural languages such as English, which are deeply intertwined with the lives of human communities and laden with centuries of history.

虽然华兹华斯在序言中直接批评的是奥古斯都王朝高度正式的诗歌措辞，但他对“任意”一词的使用使他参与了 18 世纪关于语言本质的更广泛的辩论。像孔多塞侯爵这样的洛克主义者认为，语言可以而且应该被改变，以更好地反映对世界的理性理解。对华兹华斯来说，一个重要的例子是玛丽亚·埃奇沃思 1798 年出版的《实用教育》，其中包括她父亲理查德·洛弗尔·埃奇沃思撰写的几章。77受到孔狄亚克神父和杜加尔德·斯图尔特等启蒙哲学家的启发，玛丽亚·埃奇沃思主张在日常情况下使用化学和其他科学领域的“技术术语”，从而使儿童能够深刻理解其含义。78她的教学法涉及孔多塞所倡导的语言改革的温和版本：试图让学生掌握一套基于清晰定义、从头构建的“哲学词汇”。79对华兹华斯来说，当语言需要处理情感时，这种分析方法就达到了极限。人们或许可以用分析性语言进行科学探究，但却无法用它写出真正动人的诗歌。

While Wordsworth’s immediate target in the preface is the highly formal poetic diction of the Augustans, his use of the word arbitrary inculcates him in the broader eighteenth-century debates about the nature of language. Lockeans such as the Marquis de Condorcet had presumed that language could and should be altered to better reflect a rational understanding of the world. An important example for Wordsworth is Maria Edgeworth’s 1798 book Practical Education, which includes a few chapters written by her father, Richard Lovell Edgeworth.77 Inspired by Enlightenment philosophers such as the Abbé de Condillac and Dugald Stewart, Maria Edgeworth advocates using the “technical terms” of chemistry and other scientific fields in ordinary situations, thus enabling children to absorb their meanings deeply.78 Her pedagogy involves a softer version of the sort of linguistic reform that Condorcet proposed: an attempt to outfit students with a “philosophical vocabulary” built from the ground up on clear definitions.79 For Wordsworth, such analytical methods reach their limit when language has to deal with feeling. One might be able to conduct a scientific inquiry using an analytical language, but one could not write genuinely moving poetry with it.

一些学者将华兹华斯的序言视为学科专业化的论据，建议诗人和科学家应该关注人类生活的不同方面。80如果是这样，那么感觉问题就根本不属于逻辑和数学等科学学科的范围。但华兹华斯所构建的区分不仅仅是学科或流派的问题。如果诗歌的最终目的是快乐，正如华兹华斯所断言的，那并不意味着它与科学无关：“我们没有知识，也就是说，没有从对特定事实的沉思中得出的一般原则，而是由快乐建立起来的，并且只因快乐而存在于我们心中。科学家、化学家和数学家，无论他们不得不与什么样的困难和厌恶作斗争，都知道这一点并感受到这一点。” 81要在任何领域产生真正的知识，人们既需要科学的分析严谨性，也需要诗歌的乐趣。

Some scholars have taken Wordsworth’s preface as an argument for disciplinary specialization, a proposal that poets and scientists should concern themselves with different aspects of human life.80 If so, then matters of feeling would simply fall outside the scope of scientific disciplines such as logic and mathematics. But the distinction Wordsworth is constructing is not simply a matter of discipline or genre. If poetry’s end is pleasure, as Wordsworth asserts, that does not mean it is irrelevant to science: “We have no knowledge, that is, no general principles drawn from the contemplation of particular facts, but what has been built up by pleasure, and exists in us by pleasure alone. The Man of Science, the Chemist and Mathematician, whatever difficulties and disgusts they may have had to struggle with, know this and feel this.”81 To produce genuine knowledge in any domain, one needs both the analytical rigor of science and the pleasure of poetry.

这种对平衡的呼吁导致的后果之一是科学退出了公开的社会行动。华兹华斯在其自传体诗歌《序曲》中将几何学描述为“一个独立的世界/由纯粹的智慧创造”。82他将这个“独立的世界”视为对道德和政治的一种逃避；华兹华斯先是对法国大革命感到幻灭，随后又对戈德温式的激进主义感到幻灭，“绝望中放弃了道德问题，/而对于我未来的研究，作为我探究能力的唯一用途，/转向数学，以及它们清晰的/而坚实的证据。” 83华兹华斯和孔多塞之间可以说存在着传记上的相似之处，因为他们都在 18 世纪 90 年代中期的政治动荡中寻求数学的慰藉。但孔多塞当然没有“放弃道德问题”——相反，他将数学视为一种确定道德的手段。华兹华斯在《序曲》的后期讲述了他对科学进入道德领域的幻灭，并表示他已经学会了“保持/两种本性——/一种感觉，另一种观察——的完全分离。” 84

One ramification of this call for balance was a withdrawal of science from overt social action. In his autobiographical poem The Prelude, Wordsworth characterizes geometry as “an independent world / Created out of pure intelligence.”82 He treats this “independent world” as an escape from both morality and politics; having become disillusioned first with the French Revolution and then with Godwinian radicalism, Wordsworth “Yielded up moral questions in despair, / And for my future studies, as the sole / Employment of the enquiring faculty, / Turned toward mathematics, and their clear / And solid evidence.”83 There is arguably a biographical parallel between Wordsworth and Condorcet here, as both sought comfort in mathematics during the political turmoil of the mid-1790s. But Condorcet certainly had not “Yielded up moral questions”—instead, he had viewed mathematics as a means of pinning morality down. Wordsworth narrates his disillusionment with such incursions of science into the moral realm in the later books of The Prelude, stating that he has learned “to keep / In wholesome separation the two natures— / The one that feels, the other that observes.”84

华兹华斯关于数学的评论，无论是否直接受到他的直接影响，都表明了十九世纪该领域的发展方向。纯数学不再以物理现实为基础；在本世纪，它被视为描述我们之外的其他世界的一种方式。逻辑领域也放弃了让社会真正变得更有逻辑性的雄心，转而只关注推测性知识。理查德·沃特利在 1826 年出版的《逻辑要素》中指出，以前的逻辑学家之所以不科学，是因为他们旨在教授一般推理，而学科界限却很模糊。85沃特利通过将逻辑限制于处理三段论的形式结构而不是实际人类思维的全部复杂性来解决这种模糊性。86约翰·斯图尔特·密尔在其 1843 年出版的《逻辑体系》中也将逻辑与教育区分开来：逻辑只涉及“如何做这件事”，而不是“如何让我们自己能够做到这件事”。87科学正在退出孔多塞所设想的那种包罗万象的社会改革，而只关注事实问题；而情感的培养可以留给诗人们。

Wordsworth’s remarks about mathematics were, whether owing to his direct influence or not, indicative of the direction the field took in the nineteenth century. Pure mathematics was no longer grounded in physical reality; over the course of the century, it would come to be seen as a way of describing other worlds apart from our own. The field of logic, too, withdrew from ambitions of actually making society more logical and came to be concerned solely with speculative knowledge. In the 1826 Elements of Logic, Richard Whately argues that previous logicians have failed to be scientific because, in aiming to teach reasoning in general, they left the bounds of their discipline vague.85 Whately addresses this vagueness by limiting logic to dealing with formal structures of syllogisms, not with the full complexity of actual human thought.86 John Stuart Mill similarly separates logic from education in his 1843 System of Logic: logic deals only with “how to do the thing,” not with “how to make ourselves capable of doing it.”87 Science was withdrawing from the sort of all-encompassing social reform that Condorcet had envisioned and coming to be concerned only with matters of fact; the cultivation of feeling could be left to the poets.

布尔的体系在符号本质方面与这些早期逻辑学有所不同，但它对逻辑学应有的狭隘理解却与这些早期逻辑学如出一辙。在《思维规律研究》的最后一段，布尔用与华兹华斯关于科学与情感的二分法颇为相符的术语来划定其学科的界限：

Boole’s system differs from these earlier logics in its symbolic nature, but it follows them in this narrow conception of what logic should be. In the final paragraph of An Investigation of the Laws of Thought, Boole draws the bounds of his discipline in terms quite in line with Wordsworth’s dichotomy between science and feeling:

如果心灵在其形式推理的能力中，无论是有意识还是无意识地遵循数学定律，它就会通过其情感和行动的其他能力，通过其对美和道德健全的感知，通过其深层的情感和感情，与不同秩序的事物保持联系……如果我们把生活的兴趣和责任纳入我们的考察范围，那么任何单纯的推理过程都无法让我们理解更重要的问题。他们所呈现的！因此，正如数学或演绎能力的培养是智力训练的一部分一样，它也只是智力训练的一部分。88

If the mind, in its capacity of formal reasoning, obeys, whether consciously or unconsciously, mathematical laws, it claims through its other capacities of sentiment and action, through its perceptions of beauty and of moral fitness, through its deep springs of emotion and affection, to hold relation to a different order of things. . . . And if we embrace in our survey the interests and duties of life, how little do any processes of mere ratiocination enable us to comprehend the weightier questions which they present! As truly, therefore, as the cultivation of the mathematical or deductive faculty is a part of intellectual discipline, so truly is it only a part.88

这里“情感与行动”的搭配尤其引人注目。布尔的心灵解剖学有效地去除了数学对法国分析家们而言所蕴含的明显政治内涵，将其归类为众多学科之一，并将其​​与那些真正激发人类行为的情感因素区分开来。

The pairing of “sentiment and action” here is especially telling. Boole’s anatomy of the mind effectively drains mathematics of the overt political content that it had had for the French analysts, slotting it away as one discipline among many and distinguishing it, in particular, from the emotional factors that motivate actual human behavior.

虽然这段话可能被理解为对传统虔诚的简单致敬，但布尔发表的演讲表明，他真诚地信奉浪漫主义的文化理念。在1847年的一次演讲中，他批评了那些认为物理科学能够“在社会中实现某种智力再生”的人。89他建议，应该走出城市，走进乡村，去“培养对自然的热爱”。90布尔写道，欣赏自然之美可以矫正那些变得“过于做作”、过于执着于“社会传统精致”的人。91与华兹华斯一样，布尔打破了启蒙运动时期强调人类理性的自然观；相反，他将“自然”与荒野和乡村生活联系在一起，这些是应对城市时尚变化的良方。在布尔看来，与科学研究相比，对风景的审美欣赏是培养道德品格的更好方式。在其他地方，布尔呼应了柯尔律治早先的论点，认为经验方法只能产生“简单的事实集合”，除非它们受到更高目标的指导。92对布尔来说，科学是文化的重要组成部分，但它不能取代诗歌最适合规范的道德情感。

While this passage might be taken as a mere nod to traditional pieties, Boole’s published lectures suggest that he was sincere in his adherence to Romantic ideas of culture. In an 1847 address, he criticizes those who believe that the physical sciences are fit “to effect a species of intellectual regeneration in society.”89 He recommends, instead, “cultivating the love of Nature” by going out of the city into the countryside.90 Appreciating natural beauty, Boole writes, provides a corrective for those who have become “too artificial,” too attached to “the conventional refinements of society.”91 Like Wordsworth, Boole breaks with Enlightenment conceptions of nature that emphasize human reason; instead, he associates “Nature” specifically with the wilderness and rustic life, which present an antidote to the changing fashions of the city. In Boole’s view, an aesthetic appreciation of landscapes is a better way of developing moral character than scientific study. Elsewhere Boole maintains, echoing an earlier argument by Coleridge, that empirical methods can only produce “a mere collection of facts” unless they are guided by a higher purpose.92 Science is an important part of culture for Boole, but it cannot substitute for the moral feelings that poetry is best suited to regulate.

科学必须与其他形式的培养保持平衡的思想在十九世纪中叶广为流传。约翰·斯图尔特·密尔在 1838 年和 1840 年发表的两篇文章中，将诗人柯尔律治和功利主义哲学家杰里米·边沁视为英国社会的两股对立力量。93密尔认为，这两位思想家都看到了对方看不到的东西。边沁的盲点与文化有关：他既不能理解“自我教育”对于培养道德品质的必要性，也不能理解不同国家之间“民族性格”的差异。94密尔告诉我们，柯尔律治对这些文化问题有更深的理解，也更能欣赏过去经验中的宝贵之处，但他却没有看到既有制度中的错误。虽然密尔明确地站在边沁主义者的经验主义一边，反对柯尔律治学派的唯心主义，但他同时主张两种观点的价值，并得出结论：“这两种人，他们似乎、并且相信自己是，敌人，实际上是盟友”，他们的力量形成了“一股巨大进步力量的对立两极” 。95

The idea that science had to be balanced with other forms of cultivation was widespread in the mid-nineteenth century. In a pair of essays published in 1838 and 1840, John Stuart Mill cast the poet Coleridge and the utilitarian philosopher Jeremy Bentham as two opposing forces in English society.93 Mill argues that each of these thinkers sees what the other fails to see. Bentham’s blind spots have to do with culture: he can wrap his head around neither the necessity of “self-education” in developing moral character nor the differences in “national character” between countries.94 Coleridge, Mill tells us, has a better understanding of these cultural matters as well as a better appreciation for what is valuable in the experiences of the past, yet he fails to see the errors in established institutions. While Mill expressly sides with the Benthamites’ empiricism over the idealism of the Coleridgeans, he asserts the value in both perspectives, concluding that “these two sorts of men, who seem to be, and believe themselves to be, enemies, are in reality allies,” their powers forming “opposite poles of one great force of progression.”95

这种平衡行为表面上令人安心：科学不会再干预人类的道德发展，就像它在18世纪90年代所做的那样，造成了灾难性的后果。人类思维将被允许有机地发展，为各种情感和诗意洞见留有发展空间，而这些情感和洞见不会被严格的规则所束缚。然而，这种二元论的文化观念却为推理在某些特定领域变得更加机械化开辟了道路。在《逻辑体系》中，密尔提出了如下“格言”：“只要学科的性质允许推理过程以机械的方式进行而没有危险，那么语言就应该尽可能地建立在机械原则之上；反之，语言的构造应该尽可能地阻止纯粹的机械使用。” 96密尔解释说，机械化带来的“危险”在于，哲学家们急于将代数的力量扩展到其他知识领域，却可能忽视符号的实际意义。然而，在纯粹的演绎探究形式中，机械化是安全的，而且确实值得称赞。文化范畴使普遍代数免于与革命政治的联系。边沁在柯尔律治的调和下更容易被接受。

This balancing act was, on the surface, reassuring: science would not meddle in the moral development of people, as it had so disastrously done in the 1790s. Human thought would be allowed to develop organically, with room left for varieties of feeling and poetic insight that could not be reduced to strict rules. Yet this dualistic idea of culture opened the way for reasoning to become far more mechanical within certain restricted domains. In System of Logic, Mill offers the following “aphorism”: “Whenever the nature of the subject permits the reasoning process to be, without danger, carried on mechanically, the language should be constructed on as mechanical principles as possible; while in contrary case, it should be so constructed that there shall be the greatest possible obstacles to a merely mechanical use of it.”96 The “danger” that mechanization poses, Mill explains, is the potential that, in their eagerness to extend the power of algebra to other areas of knowledge, philosophers will lose sight of the sensible meanings of the symbols. In purely deductive forms of inquiry, however, mechanization is safe and, indeed, laudable. The category of culture saved universal algebra from its association with revolutionary politics. Bentham went down easier when sweetened with Coleridge.

布尔在很多事情上与密尔意见相左，但他同样热衷于区分可以合法机械化的实践与不能机械化的实践。在《逻辑的数学分析》的序言中，他（错误地）引用了密尔的全部“格言”。97对布尔来说，安全地机械化逻辑意味着尊重地对待其他形式的知识：“一方面，最好地防止对符号的盲目依赖，另一方面又忽视它们的正当要求，这样做的最好办法是，每一门应用数学学科都应以应用时已知的方法的精神来处理，但要以这些方法所呈现的最佳形式来处理。” 98在使用符号方法之前，也就是说，人的头脑必须“准备好接受它们”；使用符号必须“充分理解其含义”，而不是“仅仅将其作为不具暗示性的字符，依赖权威来使用”。99一旦准备完成，符号就可以接管，逻辑就可以变得纯粹机械。

Boole disagreed with Mill on many things, but he shared this keenness to separate practices that could legitimately be mechanized from ones that could not. In the preface to The Mathematical Analysis of Logic, he (mis)quotes Mill’s “aphorism” in full.97 For Boole, mechanizing logic safely means treating other forms of knowledge with respect: “It were perhaps the best security against the danger of an unreasoning reliance upon symbols, on the one hand, and a neglect of their just claims on the other, that each subject of applied mathematics should be treated in the spirit of the methods which were known at the time when the application was made, but in the best form which those methods have assumed.”98 Before one can employ symbolic methods, that is, one’s mind must be “prepared to receive them”; symbols must be used “with a full understanding of their meaning” rather than as “mere unsuggestive characters, the use of which is suffered to rest upon authority.”99 Once this preparation is complete, the symbols can take over and logic can become purely mechanical.

这段话的措辞可能显得不真诚。“秉承”其他学科的精神来运用应用数学究竟意味着什么？当然，在某种程度上，布尔试图将背离传统做法的行为合理化，并反驳那些认为他正在用机器系统取代智能思维的指责。但他的代数逻辑确实比以往尝试更温和地处理文化问题。普适代数就是这样。与孔多塞不同，布尔没有在现在和过去之间设置任何障碍；相反，他的体系应该与它所应用的科学协同工作，因为这些科学已经发展到今天。这种差异的实际意义在他处理符号含义的方式上体现得最为明显。语言曾经是朋友，而不是敌人。

The rhetoric of this passage might come off as disingenuous. What does it really mean to use applied mathematics “in the spirit of” another discipline? Certainly, to some extent, Boole was looking to rationalize a departure from received practices and fend off the accusation that he was replacing intelligent thought with a machinelike system. But his algebraic logic really did handle culture with a softer touch than previous attempts at a universal algebra did. Unlike Condorcet, Boole places no barrier between present and past; instead, his system is supposed to work together with the science to which it is applied, as that science has developed up to the present. The practical implications of this difference are strongest in the way he handled the meanings of the symbols. For once, language was a friend, not an enemy.

严谨与精神

Rigor and Spirit

1878年，布尔的妻子玛丽·埃弗里斯特·布尔发表了一些关于她已故丈夫性格的回忆录。她写道，布尔教授语言的方式是虔诚的：“我绝不允许鼓励孩子们像孩子一样滥用语言。他会坐在婴儿的膝盖上，教他们清晰地发出第一个单词。” 100布尔教授语言的用心，部分原因是为了“让孩子们养成准确的习惯”，她推测，还因为“语言是一种共同财产，他希望打消个人拥有随意使用语言的私人权利的想法。” 101最后，这也反映了他的信念：“任何宣称相信圣经的人都应该在使用语言时非常虔诚。” 102如果这种说法正确的话，乔治·布尔是一位语言非自愿论者：他认为语言是集体创造的，不受任何个人控制。这种对语言的崇敬，使他对符号的意义有了不同于莱布尼茨和孔多塞的视角。对布尔来说，他的任务并非从代数中抹去所有词语的痕迹，而是将只有词语才能赋予代数的意义注入其中。

In 1878, Boole’s wife, Mary Everest Boole, published some reminiscences about her late husband’s character. She reports that his approach to teaching language was reverent: “I was never allowed to encourage our children in any babyish corruption of language. He would sit for a length of time with an infant on his knee, teaching it to pronounce its first words with perfect distinctness.”100 The care he took in teaching language was partly “to give the children habits of accuracy” and also because, she surmises, “language being a common property, he wished to discourage the idea of individuals having a private right to use it as they pleased.”101 Finally, it reflected his belief that “people who make any sort of profession of believing in the Bible ought to be very reverent in their use of words.”102 George Boole was, if this account is just, a linguistic involuntarist: he saw words as collective creations that existed outside the control of any individual. This reverence toward language gave him a different perspective on what symbols could do than those of Leibniz and Condorcet. For Boole, the task was not to expunge all traces of words from algebra but to infuse algebra with the meaning that can only come from words.

培养对语言的尊重是当时教育的常见部分。在萨拉·波特的《算术对话》中，学生的哥哥反复建议使英语更理性的方法，而他的母亲每次都警告他不要这样妄自尊大。他称十一和十二这两个词“野蛮”，而母亲回答说“我们必须感谢我们的撒克逊祖先”。103后来，他问“进位”是否是“一种非常愚蠢的表达方式”；他的母亲回答说“最好不要做任何创新”，重要的是这个词要定义明确。104母亲也不鼓励孩子考虑改变算术系统。有一次，他提出了十进制以外的其他计数的可能性。“同样明智的是，”母亲说，“按照你的意愿调换字母表中的字符，然后根据假设造词造句。” 105对替代系统的推测可能会让“杰出的数学家”感到高兴，但她希望儿子也能效仿“更高尚、更有用的道路”，为此，最好效仿他人的做法。106

Instilling respect for language was a common part of the pedagogy of the time. In Sarah Porter’s Conversations on Arithmetic, the pupil’s older brother repeatedly suggests ways of making the English language more rational, and in each instance his mother warns him against such presumption. He calls the words eleven and twelve “barbarous,” and she responds that “we must thank our Saxon ancestors” for them.103 Later, he asks whether “carrying” a number is not “a very silly mode of expression”; his mother responds that “it is well not to make any innovation,” and that what matters is that the term is clearly defined.104 The mother also discourages the child from thinking about changing the system of arithmetic. At one point, he raises the possibility of numerations other than base ten. “It would be just as sensible,” the mother says, “to transpose the characters of the alphabet at your arbitrary pleasure, and then to form words and sentences on the supposition.”105 Speculating about alternative systems may be pleasing to “eminent mathematicians,” but she wishes her son to follow “the higher and more useful path,” for which purpose it is best to follow the practices of others.106

这种对传统神圣性的坚持背离了十八世纪后期激进的教育理论。几十年前，玛丽亚·埃奇沃思强调在自己的头脑中建立清晰的概念，而不是遵循他人的做法；她继承了孔狄亚克神父的主张，认为最好的语言是从头开始构建的语言。对这种个人主义语言方法的抵制不仅出现在教育领域，也出现在科学思想中。在《逻辑体系》中，密尔拒绝了孔狄亚克的语言改革方案，转而支持“柯尔律治学派的……学说，即任何文化源远流长的民族的语言都是神圣的遗产，是所有时代的财产，任何一个时代都不应认为自己有权改变它。” 107博物学家亚历山大·冯·洪堡在其1845年出版的著作《宇宙》中盛赞语言的“生命力”，认为其“神秘的影响力，在自由思想的社群中萌芽，并在本土土壤中生长的地方，依然最为显著地展现出来” 。108这种对德语的公开的民族主义式赞美，标志着从弗朗西斯·培根到斯坦霍普勋爵对语言的反感的彻底终结。文字不再是观察者和现实之间的扭曲透镜，而是连接文化的重要纽带，赋予科学知识以生命。

This insistence on the sanctity of convention was a departure from the radical educational theories of the late eighteenth century. A few decades prior, Maria Edgeworth had emphasized building clear conceptions in one’s own mind over following the practices of others; following the Abbé de Condillac, she averred that the best language was one that was constructed from the ground up. The reaction against such individualistic approaches to language occurred in scientific thought as well as in education. In System of Logic, Mill rejects Condillac’s program of linguistic reform in favor of “the doctrine . . . of the Coleridge school, that the language of any people among whom culture is of an old date, is a sacred deposit, the property of all ages, and which no one age should consider itself empowered to alter.”107 In his 1845 book Kosmos, the naturalist Alexander von Humboldt praises the “animating power” of language, whose “mysterious influence still reveals itself most strikingly where it springs among free-minded communities, and attains its growth upon native soils.”108 This overtly nationalistic praise of the German language marks a break from the antipathy toward language that stretched from Francis Bacon to Lord Stanhope. Words no longer stood like a distorting lens between the observer and reality but rather provided an essential link to culture that gave life to scientific knowledge.

语言的复兴与代数的符号化转型同时发生，且往往由同一批人参与。词语或许是集体财产，但代数学家可以将其中的a、x甚至 + 视为完全任意的符号，并可随意选择其解释。虽然人们或许认为符号与词语的观念不同——传统在英语中受到重视，但在a的值方面却不重要——但事情并非如此简单。对传统的重视也适用于符号；在 1842 年的百科全书“符号与符号法”一文中，布尔的密友奥古斯都·德·摩根强调了保持数学符号连续性的重要性，因为过度创新会使旧文本变得难以阅读。109相反，任意控制也可以适用于词语。威廉·惠威尔 (William Whewell) 在 1840 年写道，在现代，技术语言是“有意构建的，有既定的目的，考虑到它的联系，并以构建一个系统为目的。” 110就符号和文字而言，传统和创新分别以集体和科学治理为后盾，都以各自的方式被视为好东西。

This rehabilitation of language occurred at just the same time—often involving the same people—as the symbolic turn in algebra. Words might be collective property, but algebraists could treat their as, xs, and even their +s as wholly arbitrary signs whose interpretations could be chosen at will. While it may be supposed that the symbols were regarded differently from words—that tradition was valued in regard to the English language but viewed as unimportant in regard to the value of a—matters were not so simple. The valuing of tradition could apply to notations, too; in an 1842 encyclopedia article for “Symbols and Notation,” Boole’s close friend Augustus De Morgan stresses the importance of maintaining continuity in mathematical symbolism, since excessive innovation renders older texts unreadable.109 Conversely, arbitrary control could apply to words. In the modern age, wrote William Whewell in 1840, technical language is “constructed intentionally, with set purpose, with a regard to its connexion, and with a view of constructing a system.”110 In regard to both symbols and words, tradition and innovation, backed respectively by collective and scientific governance, were both, in their own ways, viewed as good things.

这两种态度的共存，最好解释为科学与人类思维关系的重新配置。正如在布尔和欧姆等数学家看来，康德对理性知识和历史知识的区分使得严谨性的学科标准与思想的“培养”更加分离。语言中需要与传统联系的方面是“主观”的，它们在科学家的教育和精神生活中发挥着至关重要的作用，但与真理无关。即使对于像伯纳德·博尔扎诺这样敌视康德的数学家来说，形式上的严谨性标准也越来越优先于构成人类理解力的因素。这种划分使得 18 世纪数学的核心定义问题——定义诸如和、负或无穷小等棘手词汇的问题——变得不那么重要。一个普遍的事实是，人们越不认为语言有什么问题，就越容易忽视它。当词汇不再对科学的确定性构成威胁时，代数就可以自由地向新的方向发展。

The coexistence of these two attitudes is best explained as a reconfiguration of the relation of science to the human mind. As it was adopted by mathematicians such as Boole and Ohm, the Kantian distinction between rational and historical knowledge enabled an increased separation of disciplinary criteria of rigor from the “cultivation” of thought. The aspects of language that required a connection with tradition were “subjective” matters that played a crucial role in the education and mental life of the scientist, but that did not bear on matters of truth. Even for mathematicians hostile to Kant, such as Bernard Bolzano, formal standards of rigor were increasingly privileged over the factors that went into the formation of human understanding. This division rendered the definitional concerns that had been central to mathematics in the eighteenth century—the problem of defining such tricky words as sum, negative, or infinitesimal—less important. It is a general truth that the less people think there is something wrong with language, the more easily they can ignore it. When words no longer appeared as a threat to scientific certainty, algebra was free to develop in new directions.

布尔在《思维规律》中详细讨论了符号方法的语言含义。在解释了他写这本书的总体目的之后，他开始讨论“符号及其规律”。111他将“符号”定义为具有“固定解释”的“任意标记”，并且易于与其他符号组合，并遵循取决于它们相互解释的固定规律”。112鉴于这种任意性，他认为用单个字母替换一部分单词（表达事物或事物品质的单词）是“可以的”。113然后他试图表明这些符号遵循与代数基本定律等价的组合定律。他认为，这些术语出现的顺序没有区别；“河口的河流”与“河口的河流”意思相同。114这种等价性转化为交换律xy = yx。同样，“欧洲男人和女人”与“欧洲男人和欧洲女人”意思相同，转化成分配律z ( x + y ) = zx + zy。115布尔还展示了if、not以及其他各种连接词如何映射到代数表达式上。

Boole discusses the linguistic implications of symbolic methods in the most detail in The Laws of Thought. After an introduction explaining his overall purpose in the book, he enters into a discussion of “signs and their laws.”111 He defines a “sign” as an “arbitrary mark” with a “fixed interpretation, and susceptible of combination with other signs in subjection to fixed laws dependent upon their mutual interpretation.”112 On account of this arbitrariness, he finds it “permissible” to replace a subset of words—those that express things or qualities of things—with single letters.113 He then attempts to show that these signs follow laws of combination that are equivalent to the basic laws of algebra. The order in which such terms appear, he argues, makes no difference; “rivers that are estuaries” means the same thing as “estuaries that are rivers.”114 This equivalence translates to the commutative law, xy = yx. Likewise, “European men and women” means the same thing as “European men and European women,” which translates to the distributive law, z(x + y) = zx + zy.115 Boole also shows how if, not, and various other connecting words map onto algebraic expressions.

正如他的其他著作所表明的那样，正是由于符号和词语之间的这些类比，他将自己的体系视为类似于“哲学语言”。116他在本章末尾宣称，任何被代数化遗漏的语言元素，要么修饰其他词语的含义，要么仅仅“用于表达伴随命题表达的某种情感或感觉状态，因此不属于理解的范畴，而理解正是我们当前唯一关注的。” 117这句话表明，他所说的“哲学”是指类似华兹华斯意义上的科学——也就是说，主要关注事实，而不是感觉。逻辑代数排除了这些情感方面语言并不一定是因为它们不重要，而是因为它们与逻辑之间存在着学科界限。

It is on account of these analogies between the symbols and words, as his other writings make clear, that he viewed his system as akin to a “philosophical language.”116 Any elements of language that are left out of the algebraization, he declares at the end of the chapter, either modify the signification of other words or else serve only “to express some emotion or state of feeling accompanying the utterance of a proposition, and thus do not belong to the province of the understanding, with which alone our present concern lies.”117 By “philosophical,” this remark suggests, he means scientific in something like Wordsworth’s sense—that is, concerned primarily with facts, not feelings. The logical algebra excludes these emotional aspects of language not necessarily because they are unimportant but because they are divided from logic by a disciplinary boundary.

虽然布尔用英语短语来阐释他的“思维规律”，但他显然没有试图通过对语言进行实证分析来推导这些规律。在布尔在其他地方引用的一本1847年的著作中，语言学家罗伯特·戈登·莱瑟姆曾论证道：“语言中命题的结构并不总是与逻辑中命题的结构一致。” 118布尔同样将逻辑与特定语言的惯例区分开来。他引用了约翰·弥尔顿《失乐园》中的三段话来说明，一个短语即使违反了英语词序的通常规则，也能保持其逻辑结构：

While he illustrates his “laws of thought” with English phrases, Boole is decidedly not attempting to derive them through an empirical analysis of language. In an 1847 book that Boole cites elsewhere, the philologist Robert Gordon Latham had argued “that the structure of propositions in Language does not always coincide with the structure of propositions in Logic.”118 Boole similarly sets logic apart from the conventionalities of particular languages. He uses three quotations from John Milton’s Paradise Lost to illustrate how a phrase can retain a logical structure even as it violates the usual rules of English word order:

“天上的长子。”

“黑暗而深邃的水域不断升起。”

“光明精华的光明流溢创造。” 119

“Offspring of heaven first-born.”

“The rising world of waters dark and deep.”

“Bright effluence of bright essence increate.”119

布尔认为，诗歌的运作遵循一种不受语言惯例约束的“合法自由”，这种自由“受到思想内在规律的认可” 。120因此，布尔认为，思想规律xy = yx “实际上是在语言规律中发展起来的，语言是思想的产物和工具”，尽管“出于方便的原因”，散文作家通常遵循统一的词序。121

Poetry, according to Boole, operates with a “lawful freedom” from the conventions of the language, a freedom that is “sanctioned by the intimate laws of thought.”120 Boole thus maintains that the law of thought xy = yx “is actually developed in a law of Language, the product and the instrument of thought,” even though, “for reasons of convenience,” prose writers usually follow a uniform word order.121

这个论点值得深思。弥尔顿某些诗句的可解释性如何证明逻辑规律的存在？布尔设想的究竟是什么样的解释？在他那个时代，处理这些问题的领域是诠释学，这门解释学起源于德国，后经浪漫主义传入英国。正如布尔对弥尔顿的讨论一样，浪漫主义诠释学重视理解文本的精神，而非诸如拼写和句法等“机械”的问题。122柯尔律治的作品中就出现了这种价值等级制度的一个富有启发性的例子，包括其潜在的令人不安的含义。在《方法论原则》中，柯尔律治设想了一个文盲试图理解圣经的场景：

This argument is worth puzzling over. How does the interpretability of some lines from Milton demonstrate the existence of a logical law? What kind of interpretation is Boole imagining? The field that dealt with these issues in his time was hermeneutics, the science of interpretation that originated in Germany and spread to England via the Romantics. Like Boole in his discussion of Milton, Romantic hermeneutics valued perceiving the spirit of a text over such “mechanical” matters as spelling and syntax.122 An instructive example of this value hierarchy, including its potentially troubling implications, appears in Coleridge. In his “Essays on the Principles of Method,” Coleridge imagines an illiterate person attempting to make sense of a Bible:

假设在经过长时间的劳作和不尽如人意的努力之后，他开始整理，首先是那些看起来相似的段落，然后是行和词——不，他终于发现，整个作品是由有限数量的密码、字母、标记和点的重复和交换构成的，然而，在他造诣达到顶峰和极致的完美时，他使这些东西的数量比它们将同一性质的每一种不同形式，无论是有意的还是无意的，都归类为一个独立的元素。整体而言，它既无灵魂，亦无实质，是迷信的护身符，是对科学的嘲弄。123

Say that after long and dissatisfying toils, he begins to sort, first the paragraphs that appear to resemble each other, then the lines, the words—nay, that he has at length discovered that the whole is formed by the recurrence and interchanges of a limited number of cyphers, letters, marks, and points, which, however, in the very height and utmost perfection of his attainment, he makes twentyfold more numerous than they are, by classing every different form of the same character, intentional or accidental, as a separate element. And the whole is without soul or substance, a talisman of superstition, a mockery of science.123

柯勒律治继续想象一位“友好的传教士”到来，教他读书，之后“文字变得透明，他看见它们如同看不见它们” 。124关键在于，启蒙思想家所采用的分析方法无法提供一种自给自足的文本阅读方式。一种完全未受过教育的人也能理解的写作形式，就像孔多塞的通用代数所设想的那样，是不可能的；所有的解读都需要与传统有所联系。

Coleridge goes on to imagine a “friendly missionary” arriving and teaching the man to read, after which point the “words become transparent, and he sees them as though he saw them not.”124 The point is that the analytical methods employed by Enlightenment thinkers cannot provide a self-sufficient means of reading a text. A form of writing understandable by the wholly uninstructed, as Condorcet’s universal algebra was supposed to be, is impossible; all interpretation requires some link to tradition.

几个世纪以来，诠释学呈现出各种形式，但其原始形式主要是对《圣经》和其他宗教文本的解释。柯尔律治所描述的解释具体与经文有关；《圣经》之所以变得“透明”，不仅是因为语言指导，还因为读者能够感知到透过文本闪耀的神圣真理。布尔逻辑体系的理论基础同样是宗教性的。125据他的妻子玛丽·埃弗里斯特·布尔说，他相信“神圣磁力和人的神经系统之间存在直接联系”；在《思维规律》的最后一章，他认为他的代数定律在所有宗教和哲学体系中反复出现，他以此作为其权威性的证据。126在这方面，他选择清教徒诗人弥尔顿是恰当的。古老的批评家们认为，弥尔顿故意使用难懂的句法，以确保读者自愿选择虔诚的读物；《失乐园》的含义将由信仰而不是文本本身的权威来批准。127布尔脱离语言的“合法性”在于，他的体系与人们通过文本的文字所感知到的更高真理相一致。

Hermeneutics has taken on various guises over the centuries, but in its original form it was primarily about interpreting the Bible and other religious texts. The interpretation Coleridge describes has specifically to do with scripture; the Bible becomes “transparent” not just by virtue of linguistic instruction but also because of the reader’s ability to perceive the divine truth that shines through the text. Boole’s rationale for his logic system was likewise religious.125 He believed, according to his wife, Mary Everest Boole, “that there is direct contact between the Divine Magnetism and the nervous system of man”; in the last chapter of The Laws of Thought, he argues that his algebraic laws have appeared again and again in all religious and philosophical systems, which he takes as evidence of their authority.126 In this regard, his choice of the Puritan poet Milton is appropriate. It is an old critical saw that Milton used intentionally difficult syntax so as to ensure that his readers chose a pious reading by their own free will; the meaning of Paradise Lost would be ratified by faith, not by the authority of the text itself.127 What renders Boole’s break from language “lawful” is the accordance of his system with the higher truth one perceives when one sees through the words of a text.

这至少是布尔本人的观点。如果我们换成世俗的诠释学形式，这个论点就会变得更加棘手。如果文本中闪耀的并非上帝之光，而仅仅是特定文化的偏见——或者更糟的是，意识形态——那该怎么办？柯勒律治在其阅读描述中明​​确描述的殖民情境表明，声称文本透明存在风险。书面文本不仅仅是人与人沟通的渠道，也是灌输特定教义（在柯勒律治的例子中，是基督教教义）的工具。布尔的逻辑体系可能会引发类似的批评。布尔的思维定律缺乏他所认为的神圣证明，他的自由来自语言的推理并不是真正“合法的”。相反，他通过最终任意的规则来操纵符号，因此他的结果没有任何人必须接受的特殊意义。

This, at least, would be Boole taken on his own terms. If we swap in a secular form of hermeneutics, the argument becomes more problematic. What if it is not the light of God that appears to shine through the text but simply the prejudices of a particular culture—or worse, ideology? The explicitly colonial situation Coleridge describes in his description of reading indicates the danger of claiming that a text is transparent. The written text is not merely acting as a conduit through which one being can communicate with another but also as an instrument by means of which a particular doctrine—in Coleridge’s case, a Christian one—can be instilled. Boole’s logic system could inspire a similar critique. To the extent that Boole’s laws of thought lacked the divine certification he thought they had, his freedom from language was not truly “lawful.” Instead, he was manipulating symbols through rules that were ultimately arbitrary, and his results therefore had no particular meaning that anyone was obliged to accept.

在关于符号方法的讨论中，此类担忧屡见不鲜。德·摩根在1835年分两期发表的一篇关于皮科克《代数论》的评论中写道，皮科克的作品最初看起来像“被施了魔法的符号，它们在世界各地奔波寻找意义” 。128这句话经常被引用，作为符号转向所引发焦虑的一个例子；它表明，符号操作并非真正理性，沉迷于数学抽象是走向唯我论或疯狂的途径。然而，德·摩根最终还是接受了皮科克的观点，他的评论从符号转向构建的新认识论角度批判了皮科克的著作。德·摩根并非像18世纪的许多人那样抱怨符号缺乏清晰的概念定义。相反，他关心的是确保符号操作的规则建立在合理的原则之上。

Such concerns arose frequently in discussions of symbolic methods. In a review of Peacock’s Treatise on Algebra published in two installments in 1835, De Morgan writes that Peacock’s work initially appeared like “symbols bewitched, and running about the world in search of a meaning.”128 This statement is often quoted as an example of the anxieties created by the symbolic turn; symbol manipulation, it suggests, is not truly rational, and losing oneself in mathematical abstraction is a way to solipsism or madness. Yet De Morgan ultimately came around to Peacock’s position, and his review critiques the work from within the new epistemology that the symbolic turn constructed. De Morgan is not complaining that the symbols lacked clear conceptual definitions, as so many in the eighteenth century had done. His concern, instead, is to ensure that rules of their manipulation were grounded on sound principles.

德·摩根的评论——主要阐述了他自己对符号方法的看法——试图通过阐明代数的本质来解决关于负数和虚数的混淆。德·摩根认为，代数的普遍性源于“任意的惯例，通过这些惯例，常用术语的含义小于其通俗含义” 。129因此，“某些公式的选择并非源于赋予符号的任何含义，而是符号本身的定义”。130 例如，公式a + b − b = a可以被视为减法的定义，它比算术定义更具普遍性，因为它告诉我们的内容更少——具体来说，它没有告诉我们a和b代表数字还是其他什么。由于对符号 + 和 − 的解释是任意的，数学家可以“自由”地让这些符号代表任何两个关系符合规则的运算。131这正是威廉·罗恩·汉密尔顿 (William Rowan Hamilton) 很快用来指定四元数乘法规则的那种思维方式，而仅仅几年之后，布尔 (Boole) 就将其应用于逻辑；它指向了现在所谓的抽象代数的发展——这一领域在 20 世纪初由艾米·诺特 (Emmy Noether) 充分发展，研究广义代数系统的性质，这些系统可能与数字有关，也可能无关。132

De Morgan’s review—which is mainly an essay about his own opinions on symbolic methods—attempts to settle the confusion regarding negative and imaginary numbers by clarifying the nature of algebra. Algebra, De Morgan argues, gains its generality from the use of “arbitrary conventions by which terms in common use are made to signify less than their vulgar meaning implies.”129 As a result, “certain formulæ may be chosen, not as consequences of any meaning given to the symbols, but as the definitions of the symbols themselves.”130 For example, the formula a + b − b = a may be taken as a definition of subtraction that is more general than the arithmetical definition because it tells us less—specifically, it does not tell us whether a and b represent numbers or something else. Since the interpretation of the symbols + and − is arbitrary, the mathematician is “at liberty” to make those symbols stand for any two operations whose relationship corresponds to the rule.131 This is just the sort of thinking that William Rowan Hamilton would soon employ in specifying the rules for multiplying quaternions and that, just a few years later, Boole would apply to logic; it points toward the development of what is now called abstract algebra—a field which, brought to its full development by Emmy Noether in the early twentieth century, studies the properties of generalized algebra-like systems that may or may not have a connection to numbers.132

这种定义方法与洛克的观点相反，认为代数符号与词语有着根本的不同。德·摩根写道，探究公式是否可以有多种解释，似乎就像在问“两种不同的语言是否可能所有词语都相同，但含义不同，以至于我们用第一种语言写一篇天文学论文，就等于用第二种语言（totidem verbis ）写一篇音乐论文。” 133他同意，这很荒谬。但在代数中，规则不是像词义那样由“纯粹的约定俗成”决定的，而是由数学的普遍真理决定的。134因此，定义一个代数符号不同于定义一个词：“我们说，让 a ⁰ = 1；但这个词更像是‘我认为我们必须让a ^{0 = 1’，而不是‘我们完全可以自由地为}a ⁰选择一个含义，因此让它等于 1’。” 135在解释学中，词的定义是任意的，但解释是由更高的逻辑指导的，真理可以通过这种逻辑显现出来。在符号代数中，情况正好相反：定义由逻辑和真理指导，而解释完全是一个选择的问题。

This approach to definition suggests, contra the Lockean perspective, that algebraic symbols are fundamentally different from words. To ask whether formulae may have multiple interpretations, De Morgan writes, might seem like asking “whether two different languages might have all their words in common, but with different meanings, in such manner that by writing a treatise on astronomy in the first language, we write, totidem verbis [by all the same words], a treatise on music in the second.”133 This, he agrees, is absurd. But in algebra, the rules are determined not by “pure convention,” as the meanings of words are, but by the universal truths of mathematics.134 Defining an algebraic symbol is thus different from defining a word: “We say, let a⁰ = 1; but the word is a great deal more like ‘I think we must let a⁰ = 1’ than ‘We are perfectly free to choose a meaning for a⁰, and therefore let it be 1.’”135 In hermeneutics, the definitions of words are arbitrary, but interpretation is guided by a higher logic by virtue of which the truth can shine through. In symbolic algebra, the situation is reversed: the definitions are guided by logic and truth, whereas interpretation is wholly a matter of choice.

这种逆转的结果是，语言明显的不可控性对德·摩根来说不像对孔多塞和马塞雷斯等 18 世纪的数学家那样构成问题。词语并没有变得不重要，但德·摩根认为它们的主要意义在于教学，在教学中，词语提供了一条从熟悉的概念到高等数学复杂性的途径。在评论的结尾，德·摩根提议“写一篇关于数学中词语的使用及其与符号的联系的论文” 。136他写道，新代数“将引入一套全新的习语”，这可能会给学生带来困难；而熟练的数学家可以通过“脱离代数背景理解术语的含义”来解决这些困难。137也就是说， sum一词的传统含义可以帮助理解a + b，即使这些符号在技术上代表比普通加法更抽象的运算。这种思路与威廉·弗伦德（William Frend）在十八世纪晚期提出的论点截然相反，弗伦德认为，诸如“正方形”之类的词语的既定含义可能会通过引发虚假的联想来干扰其专业定义。对德·摩根（De Morgan）而言，词语的日常非专业含义提供了一种宝贵的“资源”，学生只要具备足够的语言能力，就可以利用这些资源来理解符号。138

A consequence of this reversal is that the apparent uncontrollability of language posed less of a problem for De Morgan than it had for eighteenth-century mathematicians such as Condorcet and Maseres. Words had not become unimportant, but De Morgan locates their relevance primarily in pedagogy, where they provide a route from familiar notions into the intricacies of advanced mathematics. Near the end of the review, De Morgan proposes “a treatise on the use of words in mathematics, and their connexion with symbols.”136 The new algebra, he writes, “will introduce a very new set of idioms” that may pose difficulties for students; an adept mathematician can navigate these difficulties by “helping himself to the meaning of the terms out of the algebraical context.”137 The conventional meaning of the word sum, that is, could aid in understanding a + b even if those symbols technically represent a more abstract operation than common addition. This line of thinking is a direct reversal of William Frend’s late eighteenth-century argument that the received meanings of words such as square might interfere with their technical definitions by calling to mind spurious associations. For De Morgan, the ordinary, nontechnical meanings of words provide a valuable “resource” that students may exploit in making sense of symbols, provided they have sufficient linguistic competency.138

正如这段话所暗示的，符号转向并没有废除词语；它只是改变了词语与符号的关系。布尔也将词语视为一种资源，一个可以从中汲取意义的源泉。词语在布尔逻辑体系中的作用常常被忽视。在编程语言中使用的形式中，布尔逻辑只有两个值：1 和 0（或者，如果我们愿意，也可以是真和假）；使用and和or 的规则，我们总是可以将布尔表达式求值为这两个值之一，例如 (0 and 1) or 1 的值为1。一些流行的说法暗示布尔的系统也是这样运作的。139但布尔的演绎程序并没有涉及用数字方式评估逻辑表达式，他也没有将他的系统限制为两个值 0 和 1。如果 1 表示一切而 0 表示无，那么布尔系统还承认无数的其他值，这些值指的是特定类别的事物，例如鸭子和水豚。这些值不对应任何数字；如果将x解释为水豚，那么符号的值就是单词的含义。系统的严谨性应该不是来自这些含义的清晰性，而是来自这些含义与逻辑运作的形式规则无关的事实。我们可能清楚地知道水豚是什么，也可能不知道，但无论如何，我们可以肯定布尔运算定律是可靠的。

As this passage suggests, the symbolic turn had not done away with words; it had merely altered their relation to symbols. Boole, too, treated words as a resource, a well from which meaning could be drawn. The role of words in Boole’s logic system has often been overlooked. In the form used in programming languages, Boolean logic has only two values, 1 and 0 (or, if we prefer, true and false); using the rules for and and or, we can always evaluate a Boolean expression to one of these two values, as (0 and 1) or 1 has the value 1. Some popular accounts have implied that Boole’s system worked like this, too.139 But Boole’s deduction procedure did not involve evaluating logical expressions numerically, nor did he restrict his system to the two values 0 and 1. If 1 means everything and 0 nothing, Boole’s system also admits an indefinite number of other values that refer to specific categories of things, such as ducks and capybaras. These values do not correspond to any numbers; if one interprets x as capybara, then the value of the symbol simply is the meaning of the word. The rigor of the system is supposed to come not from the clarity of these meanings but from the fact that the meanings are irrelevant to the formal rules by which the logic operates. We may or may not have a clear idea of what a capybara is, but either way, we can be sure that Boole’s laws of operation are solid.

形式化的转变减轻了语言的负担。除了可能出于教学目的之外，不再需要争论“负”的含义是否足够清晰，或者“和”的含义是否可以改变。在布尔的手中，形式与解释之间的区别使得通用代数能够与词语和平共处，词语可以随意替换：在一个段落中，他让y表示“好”，然后又将其含义改为“羊”。140符号并没有取代这些词语；相反，它从这些词语中，从而从英语中汲取意义。这与莱布尼茨和孔多塞处理意义的方式不同。“特征数”消失了；从感官开始构建复杂概念的尝试也消失了。甚至布尔的思维定律本身也被认为可以在“普通话语语言”中找到。141语言对布尔来说并非问题，因为他不像他的前辈那样，试图从头构建一种全新的世界观。他用对通用语言的尊重换取了对符号解释的任意控制权。

The formal turn lifted a weight from the shoulders of language. No longer was it necessary to debate whether the meaning of negative was clear enough or whether the meaning of sum could be altered, except perhaps for the purpose of teaching. In the hands of Boole, the distinction between form and interpretation made it possible for a universal algebra to coexist peacefully with words, which could be swapped out at will: in the course of a single paragraph, he makes y mean “good” and then changes its meaning to “sheep.”140 The symbol does not replace these words; instead, it draws its meaning from them, and thus from the English language. This is different from how Leibniz and Condorcet handled meaning. Gone are the “characteristic numbers”; gone, too, is the attempt to build up complex concepts from scratch, starting with the senses. Even Boole’s laws of thought themselves are supposed to be found in “the language of common discourse.”141 Language did not arise as a problem for Boole because, unlike his predecessors, he was not attempting to build a whole new way of thinking about the world from the ground up. He purchased arbitrary control over the interpretation of symbols in exchange for deference toward the common tongue.

无法解释的领域

Uninterpretable Realms

然而，布尔的体系并没有完全抛弃语言。符号要想发挥作用，就不能完全屈从于解释者的意志；它们必须能够以某种方式进行反击。不难找到一个特别有说服力的例子。在推导思维定律时，布尔考虑了“‘白人’和‘男人’所隐含的两种心理操作”。142“男人”操作选择所有属于男人的事物；然后， “白人”操作选择那些属于白人的事物。由此，人们构建了“白人”这一类。143根据布尔的说法，人们也可以颠倒顺序：首先选择“白色物体”，然后从这些物体中选择属于男人的物体。144因此，他推导出定律xy = yx 。这里的问题并不难看。他将“白人”中的白色等同于他之前的例子“有角的白羊”中的白色——也就是说，将其视为字面意义上的颜色。145布尔的体系将词语视为含义固定的，无论上下文如何，它们都表示相同的事物。这种假定的固定性使得符号可以像计数石头一样被随意使用。

Boole’s system does not, however, leave language totally alone. To be of any use, symbols cannot bend entirely to the will of the interpreter; they must somehow be able to push back. It is not hard to find a particularly charged example. In his derivation of the laws of thought, Boole considers “the two mental operations implied by ‘white’ and ‘men.’”142 The men operation selects all things that are men; the white operation then selects those who are white. One thus constructs the class of “white men.”143 One can also, according to Boole, reverse the order: one starts by selecting “white objects,” then one selects from those objects the ones that are men.144 Hence he derives the law xy = yx. The problem here is not hard to see. He is treating the white in “white men” as equivalent to the white in his earlier example, “horned white sheep”—that is, as a literal color.145 Boole’s system treats words as fixed in their meanings, signifying the same things regardless of context. This presumed fixity enables the symbols to be tossed around like counting stones.

如果拥有莱布尼茨希望创造的那种完美的分析语言，这种趋势就不会成为问题。莱布尼茨普遍特征的早期版本——即被认为能够绝对确定地解决问题的特征——需要发现一套通用的基本概念，所有其他概念都可从中构建出来；符号的意义将植根于神圣铭刻于心灵的理念，因此完全不受词语的细微之处的影响。布尔的逻辑体系，以及由此衍生的所有技术形式，并不需要如此不切实际地努力去取代语言，而危险就在于此。“白人和白羊”真的与“人和羊这样的白色物体”含义相同吗？如果不是，那么这些符号在布尔的符号化过程中产生的意义就不一定与它们最初的含义相同。

This tendency would not be a problem if one had a perfect analytical language of the sort Leibniz had hoped to create. The early version of Leibniz’s universal characteristic—the one that was supposed to resolve questions with absolute certainty—required the discovery of a universal set of basic concepts from which all other concepts could be constructed; the meanings of the symbols would be grounded in ideas divinely inscribed on the mind and thus wholly unaffected by the niceties of words. Boole’s logic system, along with all the technical forms derived from it, demands no such quixotic effort to replace language, and therein lies the danger. Does “white men and white sheep” really mean the same thing as “white objects that are men and sheep”? If not, then the symbols do not necessarily come out of Boole’s symbolic process with the same meanings they had when they went in.

布尔并非没有意识到这些问题。即使他脱离语言是“合法的”，也仍然是一种脱离——符号并不遵循日常书写中词语所遵循的规则。这正是问题的关键所在；如果这些符号除了反映英语中已有的内容之外什么也不做，那么它们就毫无价值。布尔的代数被认为构成了一种不同于英语等历史上偶然存在的语言的符号系统；正如他在1848年的文章中所述，逻辑代数类似于“人类语言的形式，如果它的规则完全建立在科学的基础上”。146孔狄亚克神父将代数与语言紧密联系起来，而布尔则在它们之间打开了一道裂痕。符号或许可以从词语中获得解释，但它们也能开启词语无法描述的新领域。

Boole was far from unaware of these issues. Even if his departure from language is “lawful,” it is still a departure—the symbols do not follow the same rules that words do in ordinary writing. That is precisely the point; if the symbols did nothing other than reflect back what was already in the English language, they would be worthless. Boole’s algebra is supposed to constitute a different sort of symbolic system from historically contingent languages such as English; as he states in the 1848 article, the logical algebra is analogous “with the forms which human speech would assume, were its rules entirely constructed upon a scientific basis.”146 Whereas the Abbé de Condillac brought algebra and language into an intimate relation, Boole opens a rift between them. The symbols may gain their interpretations from words, but they can also open new realms that words cannot describe.

布尔在《思维定律》中最著名的段落之一中直接谈到了这种分歧。他承认，将代数方法应用于逻辑的主要危险之一是，这种方法可能会产生“在它们旨在辅助的思维领域中无法解释”的表达式。147这种无法解释的陈述经常出现在布尔的文本中。例如，他的一个例子以下面的定义开头：“负责任的存在者是所有理性的存在者，他们要么自由行动，要么自愿牺牲他们的自由。” 148用符号表示，他将这个陈述表示为“ x = yz + yw ”，其中x是负责任的存在者，y是理性的存在者，z是自由行动的存在者，w是自愿牺牲自由的人。149根据布尔对乘法、加法和等式的定义，这个等式可以得到清晰的解释。然而，在分析它时，他却得出了一个毫无逻辑意义的等式：

Boole addresses this rift directly in one of the most notorious passages in The Laws of Thought. One of the prime dangers of applying algebraic methods to logic, he concedes, is that such methods might produce expressions that are “uninterpretable in that sphere of thought which they are designed to aid.”147 Such uninterpretable statements occur frequently in Boole’s text. For instance, one of his examples begins with the following definition: “Responsible beings are all rational beings who are either free to act, or have voluntarily sacrificed their freedom.”148 In symbolic notation, he renders this statement as “x = yz + yw,” with x being responsible beings, y being rational ones, z being those free to act, and w being those who have voluntarily sacrificed their freedom.149 This equation has a clear interpretation given Boole’s definitions of multiplication, addition, and equality. Yet in analyzing it, he winds up with an equation that does not have any logical meaning:

这个方程等价于普通数值代数中的x = yz + yw。150但它不具有可以通过布尔方法翻译成英语的含义，因为与其他基本代数运算符不同，除法没有逻辑对应项。151理解它的唯一方法是使用布尔程序对其进行转换，这会得到一个更长的方程，最终可以翻译成英语句子，尽管不太优雅：“非理性人包括所有负责任的人，他们要么可以自由行动，要么自愿牺牲自己的自由，因此不能自由行动；再加上不确定剩余的不负责任的人，他们没有牺牲自己的自由，因此不能自由行动。” 152

This equation is equivalent to x = yz + yw in ordinary numerical algebra.150 But it bears no meaning that can be translated into English by Boole’s method, since, unlike the other basic algebraic operators, division has no logical counterpart.151 The only way to make sense of it is to transform it using Boole’s procedure, which produces a much longer equation that can finally be translated into an English sentence, albeit a less-than-elegant one: “Irrational persons consist of all responsible beings who are either free to act, or have voluntarily sacrificed their liberty, and are not free to act; together with an indefinite remainder of irresponsible beings who have not sacrificed their liberty, and are not free to act.”152

逻辑学家从未广泛接受使用无法解释的表达式。布尔的早期拥护者之一威廉·斯坦利·杰文斯批评他的系统采用了“黑暗和符号过程”，后来的符号逻辑实践者大多试图确保人们至少在理论上始终能够理解符号的含义。153但作为采用符号方法的副作用，无法解释性值得认真对待。在布尔看来，举个例子来证明这种做法的合法性，“无法解释的符号√-1”虽然没有任何合理的含义，但可以“在三角学的中间过程中使用”。154虽然我们可能会质疑诸如此类的虚数是否真的在绝对意义上“无法解释”，但它们确实不能用通常的数量或量级概念来解释。然而，正如杰罗拉莫·卡尔达诺在十六世纪发现的那样，它们仍然可以用于计算，并产生可验证的正确数量结果。从卡尔达诺时代到十九世纪初，大多数数学家都对这类方法持怀疑态度，因为它们似乎缺乏足够的概念基础。对于布尔和约瑟夫-路易斯·拉格朗日来说，虚数表面上的不可理解性与它们作为计算工具的合法性无关；就数学有效性而言，重要的是遵循规则。

The use of uninterpretable expressions has never been widely accepted among logicians. One of Boole’s early champions, William Stanley Jevons, criticized his system for employing “dark and symbolic processes,” and later practitioners of symbolic logic mostly attempted to ensure that one could always, at least in theory, understand the meanings of the symbols.153 But uninterpretability is worth taking seriously as a side effect of the adoption of symbolic methods. As an example that, in Boole’s view, proves the legitimacy of the practice, “the uninterpretable symbol √-1,” although devoid of any sensible meaning, may be used “in the intermediate processes of trigonometry.”154 Although we might question whether imaginary numbers such as are really “uninterpretable” in an absolute sense, it is true that they cannot be interpreted in terms of ordinary notions of quantity or magnitude. Yet, as Gerolamo Cardano discovered in the sixteenth century, they can nonetheless be used in calculations that produce verifiably correct results about quantities. Most mathematicians from Cardano’s time to the early nineteenth century were suspicious of such methods because they seemed to lack an adequate conceptual foundation. For Boole, as for Joseph-Louis Lagrange, the apparent incomprehensibility of imaginary numbers has no bearing on their legitimacy as instruments of computation; all that matters, for the purposes of mathematical validity, is that one follows the rules.

陷入无法解释的胡言乱语境界的危险，正是人们拥抱形式规则的代价。布尔解释说，不可解释性是符号方法所特有的，因为“这种明显的失败过程和解释之间的对应关系并不体现在人类理性的普通应用中。” 155然而，布尔并不认为这是对代数化的打击。布尔认为，这样使用符号是有效的，“遵循建立在解释基础上的规律，但不持续参考这种解释，论证链引导我们通过无法解释的中间步骤，得到可解释的最终结果。” 156尽管布尔坚称，人们必须始终清楚地理解所用符号的含义，但这一要求实际上只在演绎过程的开始和结束时才有意义，那时需要在代数符号和其他研究领域的语言之间进行转换。在布尔的程序中，解释既没有必要，而且在涉及除法的“无法解释”的表达式的情况下，解释也是徒劳的。

The danger of wandering into a realm of uninterpretable nonsense is the price one pays for this embrace of formal rules. Uninterpretability, Boole explains, is specific to symbolic methods, since “this apparent failure of correspondency between process and interpretation does not manifest itself in the ordinary applications of human reason.”155 Boole does not, however, take this as a knock against algebraization. It is valid, Boole contends, to employ symbols “in obedience to laws founded upon their interpretation, but without any sustained reference to that interpretation, the chain of demonstration conducting us through intermediate steps which are not interpretable, to a final result which is interpretable.”156 Despite Boole’s protestations that one must always clearly understand the meanings of the symbols one uses, this requirement is only really relevant at the beginning and end of the deductive process, when it is necessary to translate between algebraic notation and the language of another field of study. In the midst of Boole’s procedure, interpretation is both needless and, in the case of “uninterpretable” expressions involving division, futile.

然而，只有使用正确的程序，才有可能从无法解释的深渊中回归。正是在这里，布尔体系的宗教基础变得重要起来。正如他在一份可能写于 1854 年的手稿中所说，因为“方法和过程确实是规律的结果，而不是任意产生的”，所以可以假定任何完整的方法都建立在“不仅仅是经验的或类比的”基础上。157代数“定律”是非任意的，这一假设使布尔能够将演绎推理变成一种机械程序，而根本不需要信奉工具主义。这个过程的步骤有时是无法解释的，这一事实可以归咎于人类理解力的有限性。在 1855 年写给约翰·彭罗斯的一封信中，布尔举了“无限的空间、永恒的时间、……完美的善良和纯洁、不变的正直和真理等等”作为无法赋予清晰明确含义的术语的例子，但人们可以通过遵循思维定律来确定地进行推理。158只要我们按照这些定律“服从”符号，我们就不可能产生任何与自然秩序不同步的东西，不管我们这些暂时的存在者多么难以理解我们计算所产生的表达。

This return from the abyss of the uninterpretable is only possible, however, if the procedures one is using are the right ones. It is here that the religious basis of Boole’s system becomes important. As he states in a manuscript probably written in 1854, because “methods and processes are truly the consequences of laws and do not spring up arbitrarily into existence,” one can presume that any complete method rests on a basis that is “not merely empirical or analogical.”157 This assumption that the “laws” of algebra are nonarbitrary enables Boole to turn deductive reasoning into a mechanical procedure without at all embracing instrumentalism. The fact that the steps of this procedure were sometimes uninterpretable could be brushed off as resulting from the finitude of human understanding. In an 1855 letter to John Penrose, Boole gives “infinite space, eternal duration, . . . perfect goodness and purity, unchanging rectitude and truth etc” as examples of terms that cannot be given clear and distinct meanings but that one can nonetheless reason about with certainty by following the laws of thought.158 So long as we use symbols in “obedience” with these laws, we cannot produce anything that is out of sync with the order of nature, however little we mere temporal beings may be able to understand the expressions our reckoning produces.

这些解释带有明显的康德思想印记，尤其是唯心论者将纯粹理性置于理解之上的思想。布尔在19世纪40年代读过康德《纯粹理性批判》的德文原版，他的作品中多次引用了康德式的观点，认为逻辑和数学只处理经验必须满足的条件，而不是任何特定经验的感官内容。159他也在他的诗歌中表达了这种观点。丹尼尔·J·科恩指出了布尔十四行诗《致数字三》中康德主义的影响：

These explanations bear the distinct markings of Kantian thought and, in particular, of the idealists’ elevation of pure reason over understanding. Boole read Kant’s Critique of Pure Reason in the original German in the 1840s, and his work contains numerous Kantian-sounding references to the idea that logic and mathematics deal only with the conditions that experience must meet rather than with the sensory content of any particular experience.159 He also expressed such views in his poetry. Daniel J. Cohen has pointed out the Kantian influence in Boole’s sonnet “To the Number Three”:

这首写于19世纪40年代末的十四行诗，既是一首赞美人类智慧的赞歌，也是一首赞美亚伯拉罕之神的赞歌。地上的波浪转瞬即逝，而布尔则将“那座高耸入云、深邃圆球的拱门”与无限和永恒联系起来。我们理解这些事物的能力并非源于有限的感官，而是源于“纯粹的思想”，通过这种思想，我们可以凝视存在于“现象面纱背后”的“多元系统”。在最后一对诗中，布尔坚信，我们的感官和“纯粹的思想”能力将相互协调，并与天启宗教相协调。诗歌暗示，数学之所以拥有如此独特的力量，是因为它“仅凭思想”运作，不受经验主义的影响；正因如此，它才能引导我们准确无误地获得关于数量和神性的真理。

This sonnet, written in the late 1840s, is as much a hymn to the human intellect as to the Abrahamic God. Whereas the terrestrial wave is transient, Boole identifies “yon excelling arch with orbs deep-fraught” with the infinite and eternal. Our ability to grasp such things lies not in our finite senses but in “pure thought,” through which we may gaze at “systems manifold” that exist “behind the veil phenomenal.” In the final couplet, Boole asserts his confidence that our senses and our capacity for “pure thought” will harmonize both with each other and with revealed religion. Mathematics gains its peculiar power, the poem suggests, from the fact that it works “by thought alone,” untainted by the influence of the empirical; as such, it can lead us infallibly to truths about quantity and the divine nature alike.

符号方法与“智性凝视”之间的这种关联，其悖论在于——这一悖论在十九世纪曾引起广泛反响——形式化在将数学提升到纯粹理性的空灵境界的同时，也加深了其对物理辅助的依赖。即便数学被认为独立于经验知识而存在，它也比以往任何时候都更加依赖感官：符号由纸上的记号组成，用欧姆的表述，这构成了代数的“客观”层面。数学理性正从大脑走向纸面。很快，布尔所描述的“纯粹思维”的逻辑过程也将由机器来执行。康德将纯粹理性与感官经验区分开来，最终使逻辑从人类思维走向了物理领域，这堪称思想史上的一大讽刺。

What is paradoxical in this association of symbolic methods with the “intellectual gaze”—a paradox that echoed far and wide in the nineteenth century—is that, as much as formalization elevated mathematics to the ethereal realm of pure reason, it also deepened its dependence on physical aids. Even if mathematics was supposed to exist apart from empirical knowledge, it involved the senses more than ever: the symbols consisted of marks on a sheet of paper, which, in Ohm’s formulation, constituted the “objective” aspect of algebra. Mathematical rationality was moving from the brain to the page. Soon enough, the logical processes that Boole characterized as “pure thought” would be enacted by machines as well. It is one of the great ironies of intellectual history that, by delineating pure reason from sensory experience, Kant ultimately enabled logic to move out of the human mind and into the physical realm.

布尔本人对机械化思维没什么兴趣。他知道与他同时代的查尔斯·巴贝奇，但他似乎巴贝奇在函数论方面的工作比他的计算机引擎计划更令人印象深刻。161然而，布尔系统对机械化的适应性在他的教学方法中表现得很明显。根据玛丽布尔的说法，他认为孩子们“应该花大量时间在一些可以在没有老师的情况下完成的机械工作上，他们必须集中全部精力，并且做得非常准确。” 162因此，在向学生解释规则之前，必须教会他们如何机械地计算总和；他们“先服从，后理解”。163这与莎拉波特的洛克式教学法形成了鲜明的对比，后者通过解释计算规则背后的原因来引导学生了解计算规则。在符号方法摆脱了对概念的依赖之后，数学和逻辑的有效性与人们的思考方式的关系比与人们的行为的关系更大。从这种“指导方法”到逻辑推理的字面机械化只有很短的路程。

Boole himself showed little interest in mechanizing thought. He was aware of his near contemporary Charles Babbage, but he seemed to be more impressed by Babbage’s work on the theory of functions than by his plan for a computing engine.161 The amenability of Boole’s system to mechanization does, however, become apparent in his approach to pedagogy. According to Mary Boole, he thought it important that children “should spend a great deal of time over some mechanical work which could be done without the presence of a teacher, and which they must concentrate their whole energies upon, and do with perfect accuracy.”162 Students, then, must be taught how to work a sum mechanically before the rule is explained to them; they are “to obey first and understand afterwards.”163 The contrast with the Lockean pedagogy of Sarah Porter, which leads students toward the rules of computation by explaining the reasoning behind them, is plain. After symbolic methods were freed from their dependence on concepts, mathematical and logical validity came to have less to do with how one thinks than with what one does. It is only a short way from this “directive method” to the literal mechanization of logical reasoning.

布尔逻辑的发明者绝不会走这条路。虽然他的学生可能从机械工作开始，但他们最终应该达到顿悟的时刻，最终掌握规则背后的主导思想。164理解算法背后推理的需要一直存在，正如我们看到艾达·洛夫莱斯直接阐述的那样。但顿悟也可能永远不会到来，逻辑机器将继续大量地生产出符合系统标准的符号，无论人们是否理解它们。布尔系统的命运就是变成这样一台机器。原本应该用来补偿符号偶尔无法解释的神圣制裁，却在混乱中迷失了方向。由于规则本身就带有随意性，主宰我们现代世界的形式语言——最突出的是编程语言——更加清晰地区分了理解的人和必须满足于服从的人。

The inventor of Boolean logic would never have traveled down this road. While his students may have begun with mechanical work, they were eventually supposed to reach a moment of epiphany in which they finally grasped the leading idea behind the rules.164 The need to appreciate the reasoning behind algorithms, which we have seen Ada Lovelace state directly, persisted. But the epiphany could just as well never come, and the machine of logic would go on cranking out symbols that are correct by the standards of the system whether one understands them or not. It was the fate of Boole’s system to become such a machine. The divine sanction that was supposed to compensate for the symbols’ occasional uninterpretability got lost in the shuffle. To the extent that their rules are, instead, merely arbitrary, the formal languages that dominate our modern world—most prominently, programming languages—sharpen the distinction between those who understand and those who must be content to obey.

形式主义与文化进步

Formalism and the Progress of Culture

根据他的一些言论，人们会认为布尔的逻辑体系根本没有实际用途。在《逻辑的数学分析》的前言中，他否认自己试图“取代普通理性的使用，或使其受制于技术形式的严格约束” 。165他告诉我们，他的目的仅仅是为思辨知识做出贡献。《思维定律》确实包含应用，但玛丽·布尔后来声称，她丈夫的意图完全是宗教性的，是“遵循摩西五经的命令”而执行的，他收录这些例子只是为了“表明他的体系并非宗教狂热的幻想产物” 。166 尽管布尔家族拥有圣公会背景，但他们却是宗教折衷主义者；他们从犹太教、早期基督教和印度教等各种传统中寻求精神洞见。如果我们相信这一点，乔治·布尔逻辑代数的主要价值在于它能够帮助我们思考神圣计划的和谐性。

Based on some of his remarks, one would think that Boole’s logic system had no practical use at all. In the preface to The Mathematical Analysis of Logic, he denies that he is out “to supersede the employment of common reason, or to subject it to the rigour of technical forms.”165 His purpose, he tells us, is only to contribute to speculative knowledge. The Laws of Thought does include applications, but Mary Boole later claimed that her husband’s intentions were entirely religious, executed in “obedience to the commands of the Pentateuch,” and that he only included these examples “to show that his system was not a mere fanciful outcome of religious fervor.”166 Although they had an Anglican background, the Booles were religious eclectics; they sought spiritual insight in various traditions including Judaism, early Christianity, and Hinduism. The main value of George Boole’s logical algebra, if we believe this, is as a way of contemplating the harmony of the divine plan.

但如今的布尔逻辑是一个技术问题，而不是宗教问题。布尔的追随者很快就开始探索他的思想为机械化带来的可能性。1869 年，杰文斯使用他的简化版布尔代数开发了一种“逻辑钢琴”，可以自动得出某些类型的逻辑结论（图 4.2 ）。167布尔逻辑还构成了另一种机器的基础，这种机器将符号逻辑变成了通用的信号处理模型。人们通常将功劳归于美国数学家 C.E. 香农，他后来被称为统计信息理论的创始人，他注意到布尔代数与某些类型的电路结构相对应。香农并不是第一个有这种见解的人；C.S. 皮尔斯在 19 世纪 90 年代也有过类似的想法，尽管他没有付诸实践。168逻辑和开关电路之间的这种类比是现代电子学，特别是计算机发展的重要一步，布尔逻辑在计算机的设计和编程中仍然发挥着普遍的作用。

But Boolean logic as it exists now is a technological matter, not a religious one. Boole’s followers wasted little time in exploring the possibilities his ideas opened for mechanization. In 1869, Jevons used his simplified version of Boolean algebra to develop a “logic piano” that could automatically draw certain types of logical conclusion (figure 4.2).167 Boolean logic also formed the basis of a different sort of machine that turned symbolic logic into a general-purpose model of signal processing. Credit typically goes to the American mathematician C. E. Shannon, later known as the founder of statistical information theory, for noticing that Boolean algebra corresponded to the structure of certain types of electrical circuits. Shannon was not the first to have this insight; C. S. Peirce had a similar idea in the 1890s, although he did not put it into practice.168 This analogy between logic and switching circuits was an important step in the development of modern electronics and, in particular, computers, in the design and programming of which Boolean logic still plays a pervasive role.

布尔的工作也激发了莱布尼茨关于符号力量的思想的复兴。大约从 1874 年开始，德国数学家恩斯特·施罗德开始将布尔和欧姆的思想扩展为代数的一般理论。169对施罗德来说，代数是对运算的研究，与任何概念解释无关。他将这些运算形式化为“算法”，尽管他的意思并不是现代的算法。用施罗德的话来说，算法（Algorithmen）是确定可逆运算性质的方程的集合；例如，方程ab = ba、a ( bc ) = ( ab ) c和 ( a / b ) b = a可用于构建乘法及其逆除法的“算法”。170算法一词的这种用法，沿袭了莱布尼茨的思想，并非指一步一步的程序，而是指一组可用于代数运算的方程式。施罗德后来将他的系统扩展为一种“密码学”，其中包含了一系列逻辑关系的符号。在此基础上，他设计了一种“绝对代数”，他写道，这将构成一种与口语根本不同的“科学通用语言” 。171

Boole’s work also inspired a revival of Leibnizian ideas about the power of symbols. Starting around 1874, the German mathematician Ernst Schröder began to extend the ideas of Boole and Ohm into a general theory of algebra.169 Algebra, for Schröder, was the study of operations, considered apart from any conceptual interpretation. He formalized these operations as “algorithms,” although his meaning is not the modern one. In Schröder’s terms, algorithms (Algorithmen) are collections of equations determining the properties of invertible operations; for instance, the equations ab = ba, a(bc) = (ab)c, and (a/b)b = a could be used to construct an “algorithm” of multiplication and its inverse, division.170 This use of the word algorithm, following Leibniz, refers not to a step-by-step procedure but to a collection of equations that may be used to manipulate operations algebraically. Schröder later expanded his system into a “pasigraphy” that included symbols for a range of logical relations. On this basis, he planned out an “absolute algebra” that would, he wrote, constitute “a scientific universal language” fundamentally different from spoken languages.171

即使它并非完全是关于现代意义上的算法，十九世纪晚期的形式主义也创造了数学与逻辑之间的交流，这在很长一段时间内都成为算法思维的丰富源泉。戈特洛布·弗雷格在其1879年的小册子《概念符号》 （Begriffsschrift ，即《概念符号》）中引入了一种新的逻辑符号，它包含几个重要的特征布尔体系所缺乏的，包括量词“存在”和“对一切”，这使得用纯符号形式化整个数学证明成为可能。弗雷格的体系并非基于未解释的符号主义——他的符号具有“概念”的一面——但它确实与施罗德一样强调形式规则是通往精确的道路。与施罗德类似，弗雷格将他的体系描述为朝着实现莱布尼茨普遍特征迈出的一步。172虽然弗雷格承认逻辑的“表意文字”不能做到莱布尼茨想要做的一切，但他仍然将符号逻辑视为长期斗争的延续反对语言——帮助哲学家们“打破文字对人类精神的统治” 。173

Even if it was not exactly about algorithms in the modern sense, late nineteenth-century formalism created an interchange between mathematics and logic that long would be a fruitful source for algorithmic thinking. In his 1879 pamphlet Begriffsschrift (Concept Notation), Gottlob Frege introduced a new logical notation that included several important features that Boole’s system lacks, including the quantifiers “there exists” and “for all,” which made it possible to formalize the entirety of a mathematical proof in purely symbolic terms. Frege’s system is not based on uninterpreted symbolism—his notation has a “conceptual” aspect—but it does share Schröder’s emphasis on formal rules as a path to precision. Similarly to Schröder, Frege described his system as a step toward the realization of Leibniz’s universal characteristic.172 While Frege concedes that a logical “ideography” cannot do everything that Leibniz had intended, he nonetheless frames symbolic logic as a continuation of the long-standing struggle against language—an aid to philosophers in their quest “to break the domination of the word over the human spirit.”173

正如这句话所表明的，培根对文字的旧敌意在 19 世纪后期强势回归。这种态度将弗雷格和其他 20 世纪后期形式主义者与布尔区别开来，后者更感兴趣的是让符号与语言的主观方面协同工作。174但这种差异的程度不应被夸大。布尔有时被指责为弗雷格所说的“心理主义”，意思是将逻辑规则与关于心灵的经验事实混为一谈。175但正如布尔在《思想的规律》最后一章中所阐明的，他并不认为思想规律是像物理定律那样的经验命题；它们应该是人们在不同情况下可能或多或少忠实遵循的一般约束。176布尔和弗雷格一样，将逻辑与实际的人类思想区分开来；19 世纪后期发生变化的是形式主义者赋予思想发展的历史过程的价值。

As this statement indicates, the old Baconian hostility toward words returned in force in the late nineteenth century. This attitude differentiates Frege and other late-century formalists from Boole, who was more interested in making symbols work together with the subjective aspects of language.174 Yet the extent of this difference should not be overstated. Boole is sometimes charged with what Frege called “psychologism,” meaning conflating the rules of logic with empirical facts about the mind.175 But as Boole makes clear in the final chapter of The Laws of Thought, he does not regard the laws of thought as empirical propositions like the laws of physics; they are supposed to be general strictures that people may follow more or less faithfully in different situations.176 Boole distinguished logic from actual human thought just as much as Frege did; what changed in the late nineteenth century was the value formalists assigned to the historical processes by which thought developed.

对布尔来说，人类思想的进步受制于普遍的、神圣的法则，这些法则引导它朝着一个目标前进。他在《思想的法则》最后一章中写道：“只要人类有进步，只要社会不是停滞不前或加速衰败，道德能力就会倾向于赞同并遵循既表明永久性又表明有规律的方式从事某类行为。” 177他继续论证说，他的代数“定律”在不同宗教、文化和历史中反复出现。178正因为这种普遍趋势，这些定律才被认为是普遍的，即使它们在特定的历史和文化背景以及特定的个体中发展到不同的程度。

For Boole, the progress of human thought is subject to universal, divinely sanctioned laws that direct it toward a single goal. The moral faculty, he writes in the last chapter of The Laws of Thought, “tends, wherever human progress is observable, wherever society is not either stationary or hastening to decay, to attach itself to certain classes of actions, consentaneously, and after a manner indicative both of permanency and of law.”177 He goes on to argue that his algebraic “laws” have occurred again and again across religions and cultures and throughout history.178 The laws are supposed to be universal because of this general tendency, even if they have developed to varying degrees in particular historical and cultural contexts and among particular individuals.

这种逻辑“定律”的概念与布尔的宗教观点密不可分。与他那个时代的其他代数学家一样，布尔从神的天意角度理解数学史。威廉·罗恩·汉密尔顿 (William Rowan Hamilton) 写道“语言学”代数学派，其中包括皮科克、欧姆和格雷戈里——这些思想家都将符号方法与对语言的深入关注结合在一起。179皮科克在《大都会百科全书》中关于“算术”的文章中，对史前时期算术的发展进行了广泛的研究，以弗里德里希·施莱格尔和威廉·冯·洪堡等语言学家的工作为基础。180虽然皮科克考虑了各种文化之间的差异，但他显然认为印地语 - 阿拉伯语体系是所有进步自然趋向的目标。布尔同样将文化差异简化为朝着单一真理前进的水平。在题为“知识文化的社会层面”的演讲中，布尔引用了惠威尔的史学认为，进步是由“社会国家缓慢但共同的行动”推动的，每一项新的进步都依赖于过去在科学和艺术领域所取得的一切成就。181因此，科学知识的形成依赖于文化，并且是文化的一部分，然而，它却受到神圣天意的引导，朝着正确的理论发展。

This conception of a logical “law” cannot be disentangled from Boole’s religious views. Like other algebraists in his time, Boole understood the history of mathematics in terms of divine providence. William Rowan Hamilton wrote of a “Philological” school of algebra, in which he includes Peacock, Ohm, and Gregory—thinkers who all combined symbolic methods with a deep attention to language.179 In his article on “Arithmetic” in the Encyclopædia Metropolitana, Peacock presents extensive research about the development of arithmetic from prehistoric times, building on the work of philologists such as Friedrich Schlegel and Wilhelm von Humboldt.180 While he considers the differences between a variety of cultures, Peacock clearly thinks the Hindi–Arabic system is the one goal toward which all progress naturally tends. Boole likewise reduces cultural differences to levels of progress toward a single truth. In a speech entitled “The Social Aspect of Intellectual Culture,” Boole argues, drawing on Whewell’s historiography, that progress is driven by “the slow but combined action of the social state,” with each new advance depending on everything that has been achieved in the past in both the sciences and the arts.181 The formation of scientific knowledge, then, depends on and is a part of culture, and yet it is guided by divine providence toward the correct theory.

这种目的论是维多利亚时代早期科学与历史交叉的独特体现。到了施罗德和弗雷格的时代，关于数学进步的讨论转向了对多样性的更多认可——即不同国家实践方式的差异。在某些情况下，这种认可伴随着种族或民族的刻板印象。例如，WW劳斯鲍尔 (W. W. Rouse Ball) 在其 1889 年出版的《剑桥数学研究史》中推测了“种族在数学方法选择方面的影响”。182鲍尔声称“闪米特人种在算术和代数方面有特殊的天赋”，而希腊人更适合几何；他认为英国人特别适合分析方法，尽管牛顿主义者压制了他们一个多世纪。183这意味着，根据当时的种族理论，某些人群天生就对某些类型的数学比其他类型的数学更感兴趣。

This teleology is distinctive of how science and history intersected in the early Victorian period. By the time of Schröder and Frege, discussions of mathematical progress had turned toward a greater recognition of multiplicity—of the variation in practices from country to country. In some cases, this recognition was attended by racial or ethnic stereotypes. In his 1889 History of the Study of Mathematics at Cambridge, for instance, W. W. Rouse Ball speculates about “the influence of race in the selection of mathematical methods.”182 “The Semitic races,” Ball claims, “had a special genius for arithmetic and algebra,” whereas the Greeks were better suited for geometry; he thinks that the English are especially fitted for analytical methods, even though Newtonians suppressed them for more than a century.183 The implication is that some groups of people, as classified by the racial theories of the time, are inherently attracted to certain types of mathematics over others.

还值得一提的是，魏尔斯特拉斯在给一名学生的私人信件中经常被引用的关于数学家和诗人的评论，是在反犹太主义的背景下出现的。魏尔斯特拉斯声称“闪米特部落”的成员（特别是犹太数学家利奥波德·克罗内克）缺乏成为真正数学家所需的诗意想象力。184这种种族主义区分触及了19世纪文化观念中隐含的一种矛盾：如何调和数学表面上的普遍性与它在不同人群中发展不同的事实？在当时的思维方式下，一个显而易见的答案是将不同群体的人相互对立。185当然，并非所有数学家都持有这种观点；然而，魏尔斯特拉斯的评论表明，将纯数学提升到计算等“纯粹机械”活动之上的关注如何反映和强化了当时的偏见。

It is also worth mentioning that Weierstrass’s oft-quoted remark about mathematicians and poets, which appeared in a private letter to one of his students, occurred in an anti-Semitic context. Weierstrass claims that members of “the Semitic tribes” (and, specifically, the Jewish mathematician Leopold Kronecker) lack the poetic imagination needed to become true mathematicians.184 Such racist distinctions addressed a tension that was implicit in the nineteenth-century idea of culture: how can one reconcile the apparent universality of mathematics with the fact that it develops differently among different groups of people? To the mindset of the time, an obvious answer was to rank groups of people against each other.185 Certainly, not all mathematicians held such views; yet Weierstrass’s remark shows how the concern with elevating pure mathematics above such “merely mechanical” activities as computation could reflect and reinforce the prejudices of the time.

19世纪末，人们对纯数学本质的看法发生了变化，尽管结果难以达成共识。19世纪60年代，魏尔斯特拉斯首次对实数进行了现代描述，他基于数轴的连续性而非计数或测量的概念来解释实数。从19世纪70年代开始，格奥尔格·康托尔引入了集合论的雏形，为推理无穷提供了一种新的方法。186 1888年和1889年，理查德·戴德金和朱塞佩·皮亚诺整理出了后来成为标准公理化的东西算术。187这些以及其他十九世纪晚期数学家的工作暗示，数字比之前设想的更加奇特和违反直觉：例如，康托尔证明了实数（被认为具有无限小数展开式）是不可数的，这意味着任何试图按线性顺序列出它们的尝试总会遗漏一些数字。新的数字定义也引发了关于数学实体形而上学的棘手问题，这促使克罗内克（康托尔著作的批评者）宣称：“上帝创造了整数，其他一切都是人类的杰作。” 188

Opinions about the nature of pure mathematics were changing in the late nineteenth century, although the result was hardly a consensus. In the 1860s, Weierstrass gave the first modern account of the real numbers, which he explained based on the continuity of the number line rather than notions of counting or measurement. Starting in the 1870s, Georg Cantor introduced the rudiments of set theory, which provided a new way of reasoning about the infinite.186 In 1888 and ’89, Richard Dedekind and Giuseppe Peano assembled what would become the standard axiomatization of arithmetic.187 The work of these and other late nineteenth-century mathematicians implied that numbers were stranger and more counterintuitive than before suspected: Cantor, for instance, proved that the real numbers, understood to have infinite decimal expansions, are uncountable, meaning that any attempt to list them in a linear order will always leave some numbers out. The new definitions of number also raised thorny questions about the metaphysics of mathematical entities, inspiring Kronecker (a critic of Cantor’s work) to state that “God made the integers, all else is the work of man.”188

魏尔斯特拉斯及其同时代人发展的实数系统加深了代数在数学和通常的数字理解之间造成的差距。三个世纪前，西蒙·斯蒂文引入十进制表示法来挑战经典的几何数论，而将数字建立在数学从业者的日常计算技术之上。相比之下，在新系统中，实数与几何和实际计算都保持距离。康托尔定理意味着存在一些数字，这些数字完全无法通过任何类似印地语-阿拉伯算法的程序获得：实际上是随机数字的字符串，它们不遵循任何可以描述的逻辑而延伸到无穷大。戴德金用现在称为戴德金分割的方法解释了无理数，即将有理数分为两个集合，其元素无限接近于无理数在数轴上的位置。康托尔和戴德金的理论可以说是反算法的，因为他们假定存在无法通过任何有限程序构造的实体。数学史学家杰里米·格雷因此将戴德金与克罗内克进行了对比，后者提出了一种将计算方法置于中心的另一种数字观点。189然而，康托尔的工作提出了一个哲学问题，这个问题在艾伦·图灵等后来的思想家手中，激发了数学计算理论的发展：如果不是所有的数字都可以计算，那么哪些可以？因此，正如我们已经在布尔和洛夫莱斯身上看到的那样，机械程序在数学中的作用被降级为有限——这种程序无法提供“完整”数学家所需的诗意洞察力，正如韦尔斯特拉斯所说——最终刺激了算法理论的发展。

The real number system developed by Weierstrass and his contemporaries deepened the gap that algebra had created between mathematical and common understandings of number. Three centuries before, Simon Stevin had introduced decimal notation as a challenge to classical, geometric number theories, founding number instead on the workaday computational techniques of the mathematical practitioners. In the new system, by contrast, real numbers stood aloof from geometry and practical computation alike. Cantor’s theorem implied the existence of numbers that were utterly inaccessible by any procedure resembling the Hindi–Arabic algorism: strings of effectively random digits that extended to infinity without following any logic that could be described. Dedekind explained the irrationals by what is now called the Dedekind cut, a division of the rational numbers into two sets whose elements infinitely approach the position of the irrational on the number line. The theories of Cantor and Dedekind were arguably antialgorithmic in that they posited the existence of entities that could not be constructed through any finite procedure. The historian of mathematics Jeremy Gray thus contrasts Dedekind to Kronecker, who proposed an alternative view of number that placed computational methods at the center.189 Yet Cantor’s work opened a philosophical question that would, in the hands of later thinkers such as Alan Turing, motivate the development of a mathematical theory of computation: if not all numbers can be computed, which ones can? Thus, as we have already seen in Boole and Lovelace, the relegation of mechanical procedures to a limited role within mathematics—the idea that such procedures cannot provide the poetic insight needed by a “complete” mathematician, as Weierstrass had put it—ultimately spurred on the development of what would eventually become the theory of algorithms.

这些新的数字概念将微积分的旧几何解释边缘化，尤其是教学领域。戴德金在其1872年出版的小册子《连续性与无理数》的前言中，惊愕地承认自己在教授微积分时“求助于几何证明”。190他继续说道，几何直觉是“从教学角度来看，这极其有用，而且如果不想浪费太多时间，它确实必不可少。但这种引入微积分的方式并不能自称是科学的，这一点无人否认。” 191这就是主客体之分的简单概括：使一个系统为人所理解与使其科学化是两码事。这种区分在今天的数学课程中仍然根深蒂固：大学微积分通常采用非正式的解释和视觉辅助，而真正的科学基础则在数学分析中教授，这通常是一门针对专业学生的高级课程。

These new conceptions of number relegated the old geometric explanation of calculus to the margins and, in particular, to the realm of pedagogy. In the preface to his 1872 pamphlet Continuity and Irrational Numbers, Dedekind admits with consternation to having “recourse to geometric evidences” when teaching calculus.190 Geometric intuition, he continues, is “exceedingly useful, from the didactic standpoint, and indeed indispensable, if one does not wish to lose too much time. But that this form of introduction into the differential calculus can make no claim to being scientific, no one will deny.”191 Here, then, is the subject–object divide in plain form: making a system comprehensible to people is an entirely separate matter from making it scientific. This division remains encoded in the mathematical curriculum today: college calculus usually employs informal explanations and visual aids, whereas the truly scientific foundation is taught in Mathematical Analysis, typically a senior-level course aimed at majors.

这种分裂的后果是数学和逻辑系统与交流实用性之间出现了新的绝对疏远。尽管弗雷格将他的系统构建为莱布尼茨普遍特征的继承者，但他只关注莱布尼茨整个计划的一部分——只关注符号在逻辑推理中的使用，而没有关注交流，就交流而言，他的符号在任何意义上都不是真正通用的。莱布尼茨项目的交流方面在逻辑上的影响不如在世界语等国际辅助语言 (IAL) 的发展中的影响大。世界语由 L.L. 柴门霍夫于 1887 年首次引入，是一种口语和书面语，具有简单、一致的语法，旨在易于学习，以促进国际交流。192莱布尼茨的工作中也有此类项目的先例，就像符号逻辑一样；莱布尼茨的项目之一就是拉丁语的简化版。这个想法在20世纪成为了几种人工语言（IAL）尝试的基础，其中一种是由数学家朱塞佩·皮亚诺（Giuseppe Peano）提出的。然而，皮亚诺明确表示，他认为这种人工语言与数学符号完全不同。193到了20世纪初，两种旨在实现不同通用性的人工语言——用于科学目的的数学和逻辑符号，以及旨在取代口语的IAL——已经分化。

A consequence of this division was a newly absolute alienation of mathematical and logical systems from the practicalities of communication. Although Frege framed his system as a successor to Leibniz’s universal characteristic, he focused only on a portion of the total Leibnizian program—only on the use of symbols in logical reasoning, not so much on communication, with regard to which his notation is not, in any meaningful sense, actually universal. The communicational side of Leibniz’s project came to perdure less in logic than in the development of international auxiliary languages (IALs) such as Esperanto. First introduced by L. L. Zamenhof in 1887, Esperanto is a spoken and written language with a simple, consistent grammar designed to be easy to learn so as to facilitate international communication.192 Such projects had precedents in Leibniz’s work just as much as symbolic logic did; one of Leibniz’s projects was a simplified version of Latin. This idea would, in the twentieth century, become the basis of several attempted IALs, one of them by a mathematician: the aforementioned Giuseppe Peano. Yet Peano made it clear that he saw this constructed language as entirely distinct from mathematical notation.193 By the turn of the twentieth century, two types of artificial language aiming at different sorts of universality—mathematical and logical notations meant for scientific purposes and IALs meant to replace spoken vernaculars—had diverged.

这种分歧恰逢数学和语言学科之间界限的日益清晰。在18世纪，语言是否是意识设计的对象一直是一个争论的话题，而代数的例子提供了证据。到19世纪末，语言学家们就相反的结论接近达成共识。正如安德里亚·亨德森所指出的，正如语言学家开始将形式视为可以用科学精确度描述的东西一样，他们也开始将内容——也就是意义——视为神秘的、无法被确定或控制的东西。194 19世纪30年代，语言学家威廉·冯·洪堡曾将语言称为“心灵的无意识流露”，人们使用它时“并不知道自己是如何”已经塑造了它。” 195这种非自愿主义成为索绪尔 1916 年《普通语言学教程》中的一种学说，该教程由他的学生根据他的讲座记录下来。这本书包含一个详细的论证：“一旦符号在语言共同体中确立下来，个人就无权以任何方式改变它。” 196也就是说，符号是集体管理的，（与洛克相反）人们不能简单地选择以自己想要的方式理解一个词。

This divergence coincided with a sharpening of the boundary between mathematical and linguistic disciplines. In the eighteenth century, it had been a matter for debate whether languages were objects of conscious design, and the example of algebra offered evidence that they were. By the end of the nineteenth century, linguists were nearing a consensus around the opposite conclusion. As Andrea Henderson has pointed out, just as linguists came to see form as something that could be described with scientific precision, they began to see content—that is, meaning—as mysterious, incapable of being pinned down or controlled.194 In the 1830s, the philologist Wilhelm von Humboldt had called language “an involuntary emanation of the mind” that people use “without knowing how they have fashioned it.”195 This involuntarism became a doctrine in Ferdinand de Saussure’s 1916 Course in General Linguistics, which was written down by his students based on his lectures. This book contains a detailed argument that “the individual has no power to alter a sign in any respect once it has been established in a linguistic community.”196 Signs, that is, were governed collectively, and (contra Locke) one cannot simply choose to understand a word however one wants.

索绪尔的符号理论并非当时唯一的选择，语言非自愿论也一直备受争议。1903 年，维多利亚·韦尔比夫人提出了一种名为“意符”的对比理论，旨在通过提供一套发展意义的技术，将语言“从‘本能层面’提升到意志和完全理性的层面”。197在二十世纪，改进语言的愿望成为分析哲学的核心关注点。分析哲学是一种以逻辑形式主义为基础、以发展澄清概念的方法为主的英美传统。然而，索绪尔的非自愿论在​​语言学领域占据主导地位，它有助于巩固以下观点：语言是受任何个人都无法克服的力量所支配的自然现象。语言可能遵循可以抽象陈述的规则，但语言科学的职责只是描述这些规则，而不是改变它们。

Saussure’s theory of signs was not the only option available at the time, and linguistic involuntarism never went uncontested. In 1903, Lady Victoria Welby introduced a contrasting theory called significs, which was meant to elevate language “from the ‘instinctive level’ to the volitional and fully rational plane” by providing a set of techniques for developing meaning.197 In the twentieth century, the desire to improve language would become a central concern of analytic philosophy—a predominantly Anglo-American tradition that builds upon logical formalism to develop methods of clarifying concepts. Yet Saussure’s involuntarism became dominant in the field of linguistics, where it helped solidify the idea that languages were natural phenomena governed by forces that no individual could overcome. Language may follow rules that can be stated abstractly, but the purview of linguistic science is only to describe those rules, not to change them.

数学和逻辑中使用的符号并不完全符合这种新兴的语言共识。在这些技术领域，意义由明确的、通常是书面的定义所支配，这些定义固定了符号在系统中的角色。符号学的巅峰时期是阿尔弗雷德·诺斯·怀特黑德和伯特兰·罗素合著的《数学原理》（1910-13），该书试图用尽可能简单的公理和规则来解释所有数学。他们对符号“1”的定义，让我们得以略窥这个极其复杂的系统：198

The symbols used in mathematics and logic did not clearly fit this emerging consensus about language. In these technical fields, meaning was governed by explicit, usually written definitions that fixed the symbols’ roles within systems. Peak notation occurred with Alfred North Whitehead and Bertrand Russell’s Principia Mathematica (1910–13), which attempts to demonstrate all of mathematics in terms of the simplest possible set of axioms and rules. Their definition of the symbol “1” gives a flavor of this immensely complex system:198

在第一行中，“1”被定义为所有α的类，使得对于某个x，α等于一个仅包含x的类。接下来的两行给出了该定义的等价表述。与索绪尔对符号的论述不同，罗素和怀特海将这种定义视为自愿性的，他们提出自己的体系供读者思考，并让读者自行选择是否采纳。就他们确实说服读者而言，他们所做的正是索绪尔认为不可能的事情：发挥改变符号的力量。

In the first line, “1” is defined as the class of all α such that, for some x, α is equal to a class containing only x. The next two lines offer equivalent statements of the definition. In contrast to Saussure’s account of signs, Russell and Whitehead treat this sort of definition as voluntary, presenting their system for consideration and leaving it to readers to choose whether or not to adopt it. To the extent that they did persuade, they were doing just what Saussure thought was impossible: exerting the power to alter signs.

区别在于定义的目的。罗素-怀特海体系并非完全形式化；它运用了未定义的“原始思想”，包括布尔代数的运算。199然而，它的定义仅处理符号之间的关系，并以方程的形式呈现（类似于德·摩根提出的减法定义）。与索绪尔所谴责的语言规范主义不同，这种形式定义不会干扰人类社会中既定的语义惯例；任何与符号“1”相关的先前存在的关联都被允许潜伏在背景中，帮助学生学习该体系，提供指导其使用的直觉，并简化形式符号与其他学科语言之间的翻译。这种思维方式将现代形式语言与19世纪之前的符号符号区分开来。尽管公理伪装成绝对的开端，但形式语言的目标并非与既定的思维方式进行革命性的突破。相反，它们作为学科严谨性的保证，必须与人类实践的培养（符号惯例、非正式解释和教育方法）保持平衡，使人们能够理解符号。

The difference comes down to what a definition is supposed to accomplish. The Russell-Whitehead system is not fully formal; it employs “primitive ideas” that are left undefined, including the operations of Boolean algebra.199 Yet its definitions deal only with relations among the symbols and take the shape (like De Morgan’s proposed definition of subtraction) of equations. Unlike the linguistic prescriptivism that Saussure deplores, such formal definitions do not interfere with the semantic conventions that are established in human communities; whatever preexisting associations may accompany the symbol 1 are permitted to lurk in the background, helping students learn the system, providing intuitions to guide its use, and easing translation between the formal notation and the languages of other disciplines. This way of thinking differentiates modern formal languages from what symbolic notations were before the nineteenth century. As much as axioms masquerade as an absolute beginning, formal languages do not aim at a revolutionary break with established ways of thinking. Instead, they serve as guarantors of a disciplinary rigor that must be balanced with the cultivation of human practices—notational conventions, informal explanations, and educational methods—that make the symbols comprehensible to people.

正是这些人文问题不可否认的重要性，促使弗洛里安·卡乔里（Florian Cajori）在20世纪20年代开始研究数学符号的历史。我已经引用了卡乔里1929年出版的著作作为二手资料，但现在是时候将其置于历史背景中了。卡乔里表示，他的目标是探究历史能够告诉我们什么才是好的符号。他总结了他的一个主要结论：

It was the undeniable importance of these human matters that led Florian Cajori to research the history of mathematical notations in the 1920s. I have already cited Cajori’s 1929 book as a secondary source, but the time has come to place it in its historical context. Cajori states that his goal is to determine what history can tell us about what makes for good notation. He sums up one of his main conclusions:

个人主义的失败——历史的教训清晰地表明，数学家们仍然处于巴别塔的阴影之下，个人为迅速实现符号统一所做的尝试都失败了。数学家们并没有从历史的教训中获益。他们未能采用替代方法。数学符号是供数学界使用的，因此应该被该界采用。民主的成功需要群众行动。200

Individualism a failure.—As clear as daylight is the teaching of history that mathematicians are still in the shadow of the Tower of Babel and that individual attempts toward the prompt attainment of uniformity of notations have been failures. Mathematicians have not been profiting by the teachings of history. They have failed to adopt the alternative procedure. Mathematical symbols are for the use of the mathematical community, and should therefore be adopted by that community. The success of a democracy calls for mass action.200

卡乔里接着建议，效仿1881年在巴黎召开的国际电工大会，成立一个国际委员会，该大会提出了电工计量单位的国际标准。他认为，建立统一的唯一途径是“打破极端个人主义对本质上是共产主义的问题的迷恋” 。201

Cajori goes on to recommend an international committee on the model of the International Congress of Electricians, which, in Paris in 1881, propounded international standards for electrical units of measurement. The only way to establish uniformity, he argues, is by “breaking the infatuation of extreme individualism on a matter intrinsically communistic.”201

Cajori 更倾向于有意识的规划而非传统，他表现出我们或许可以称之为现代主义的进步态度。卡乔里用一种让人联想起埃兹拉·庞德的语言，谴责了相互冲突的符号的泛滥：“哦，混沌女神，你正在侵犯最崇高的科学之一！” 202他还对数学家们不愿采用新符号感到惋惜，例如笛卡尔的指数符号，在五十多年里都没有被普遍采用。他将这种犹豫归咎于“习惯的力量”；他似乎没有想到，十七世纪的数学家们积极推崇传统，并将“创新”视为坏事。203 “在如此冗余和陈旧之中，”他总结道，“我们可以看到逝去的过去的手正紧紧抓住现在，指引着未来。” 204卡乔里的史学著作缺乏皮科克十九世纪早期算术史中那种天意般的意味，而是强调数学家们自主掌控事务、建立通用语言的实际需要。标准化机构的宗旨是将现在从过去中解放出来，用基于有意识的、集体共识的机制取代那些随意发展的惯例。

In his preference for intentional planning over tradition, Cajori exhibits attitudes toward progress that we might call modernist. In language reminiscent of Ezra Pound, Cajori deplores the proliferation of conflicting notations: “O Goddess of Chaos, thou art trespassing upon one of the noblest of the sciences!”202 He also laments the hesitancy of mathematicians to adopt new symbols, as in Descartes’s notation for exponents, which was not universally adopted for more than fifty years. He attributes this hesitancy to the “force of habit”; that seventeenth-century mathematicians actively valued tradition and viewed “innovation” as a bad thing does not seem to occur to him.203 “In such redundancy and obsoleteness,” he concludes, “one sees the hand of the dead past gripping the present and guiding the future.”204 Cajori’s historiography is devoid of the providential overtones we find in Peacock’s early nineteenth-century history of arithmetic, instead emphasizing the practical need for mathematicians to take matters into their own hands and establish a common language. Standards institutions are supposed to liberate the present from the past by replacing haphazardly developing conventions with ones based on agreements that are both intentional and collective.

走向标准化确实是二十世纪的必然趋势。类似卡乔里（Cajori）的修辞手法在计算机科学家埃兹格·迪杰斯特拉（Edsger Dijkstra）的著作中反复出现，他呼吁计算机学科摆脱过去的包袱。然而，就在卡乔里著作出版两年后，关于形式语言的讨论又发生了一次根本性的变化。1931年，库尔特·哥德尔（Kurt Gödel）不完备定理的出现，以及紧随其后的阿隆佐·丘奇（Alonzo Church）和艾伦·图灵（Alan Turing）的工作，粉碎了所有建立像罗素和怀特黑德（Russell and Whitehead）自认为已经构建的那种自给自足的数学系统的希望。哥德尔的研究结果并没有消除形式语言和自然语言之间的区别，但它们确实表明，形式规则永远不足以将系统排除在历史之外。这一发现最终不仅对纯数学，而且对一项具有重大社会意义的技术——算法——也产生了影响。

Toward standardization was, indeed, the way things would go in the twentieth century. Rhetoric much like Cajori’s would recur in the work of the computer scientist Edsger Dijkstra, who called for his discipline to shed the baggage of the past. But the conversation about formal languages underwent another fundamental change just two years after the publication of Cajori’s book. The advent of Kurt Gödel’s incompleteness theorems in 1931, with the work of Alonzo Church and Alan Turing following close behind, dashed all hopes of a self-sufficient mathematical system of the sort Russell and Whitehead thought they had constructed. Gödel’s results did not undo the distinction between formal and natural languages, but they did show that formal rules can never suffice to keep systems outside of history. This finding would ultimately have implications not just for pure mathematics but also for a technology of great social importance: the algorithm.

净化

The Purification

大约在1900年，算法通常意味着两种含义之一：一套古老的计算技术，通常在小学教授；或者一种抽象的代数结构，很少有非数学家会遇到。大约在2000年，算法已成为一种经济引擎，在新成立的谷歌服务器以及越来越多的家用电脑中得到体现。算法的概念在20世纪后期逐渐成型，与我在本书中讲述的故事有着复杂的联系。自从查尔斯·巴贝奇和艾达·洛夫莱斯的计划未能实现以来，可编程计算机的广泛使用带来了新的变化，这无疑改变了描述程序的任务。当与一台能够准确执行指令、不加质疑地执行指令的机器一起工作时，精确度就具有了新的含义。算法理论也反映了二战后时期的制度压力：竞争大学、IBM 等企业和军事机构的利益，以及持续不断的冷战背景噪音。

Circa 1900, algorithm usually meant one of two things: a venerable set of calculating techniques commonly taught in elementary school or an abstract type of algebraic structure that few nonmathematicians would ever encounter. Circa 2000, algorithms were an economic engine, enacted in the servers of the recently founded Google and in a growing number of home computers. The idea of algorithm that solidified in the late twentieth century exists in a complex relation to the story I have told in this book. One thing that was new (since the plans of Charles Babbage and Ada Lovelace had gone unfulfilled) was the widespread use of programmable computers, which certainly changed the task of describing procedures. Precision takes on a new meaning when working with a machine that does exactly what it is told, no questions asked. The theory of algorithms also came to reflect the institutional pressures of the post–World War II period: the competing interests of universities, businesses such as IBM, and military institutions, along with the persistent background noise of the Cold War.

尽管20世纪发生了诸多变化，算法理论依然展现出19世纪改变符号代数的主客体划分的持续影响。正如我在第四章中所讨论的，乔治·布尔在19世纪40年代的工作将逻辑限制在推理的形式层面，排除了人类思维形成过程中的许多其他情感和文化因素。早期的数学家试图将他们的方法建立在清晰的概念定义之上，而布尔时代的代数学家则颠倒了这些术语：套用乔治·皮科克的话来说，代数规则决定了符号的解释，而不是相反。与布尔和皮科克的代数一样，现代算法可以有多种解释方式，仅受其形式特征的限制。例如，一个典型的网络分析算法并没有说明网络代表什么；它同样适用于路线图或家庭关系图。正如形式主义将纯数学与量级和数量的物理概念分离一样，现代算法思维将程序与其应用的具体情况以及人们理解它们的概念手段区分开来——这种区分极大地促使算法成为强大而危险的力量。

But for all that changed in the twentieth century, the theory of algorithms shows the continuing influence of the subject–object divide that transformed symbolic algebra in the nineteenth. As I discussed in chapter 4, George Boole’s work of the 1840s restricted logic to the formal aspects of reasoning, excluding the many other emotional and cultural factors that went into the formation of human thought. Whereas earlier mathematicians had attempted to ground their methods on clear conceptual definitions, algebraists in Boole’s time reversed the terms: the rules of algebra, to paraphrase George Peacock, would determine the interpretation of the symbols and not the other way around. Much like algebra for Boole and Peacock, modern algorithms may be interpreted in multiple ways, constrained only by their formal qualities. A typical network analysis algorithm, for instance, says nothing about what the network represents; it could apply equally well to a road map or to a diagram of family relationships. As formalism detached pure mathematics from physical notions of magnitude and quantity, modern algorithmic thinking sets procedures apart from the concrete situations in which they are applied and the conceptual means by which people understand them—a division that contributed greatly to making algorithms the powerful and dangerous force they have become.

这种思维方式的出现很难从地理上定位。第一部关于这个主题的详细研究是 AA Markov Jr. 于 1954 年出版的《算法理论》一书，该书来自苏联。在美国政府的资助下，以色列科学翻译计划将马尔可夫的书翻译成英文；译文于 1961 年出版，尽管一些美国人在此之前就知道了马尔可夫的工作。1这种科学交流持续了几十年。1979 年，安德烈·彼得罗维奇·埃尔绍夫和唐纳德·E·克努斯组织了一次国际研讨会，让西方和苏联的计算机科学家进行了一次克努斯所说的“朝圣之旅”，前往数学家和天文学家穆罕默德·伊本·穆萨·花拉子模的故乡——乌兹别克斯坦的花拉子模地区，当时属于苏联——讨论“现代数学中的算法及其应用”。2与会者就一系列问题展开了辩论，例如如何定义算法，以及算法方法是否与代数方法有根本区别。铁幕两边的思想呈现出不同的倾向——苏联计算机科学家更多地受到马尔可夫理论的影响，而这在西方可能并不具有决定性的意义——但共同点并非毫无意义。所以算法的概念不能说是西方国家或资本主义经济体所特有的。

The emergence of this style of thinking is difficult to localize geographically. The first detailed study of the topic, A. A. Markov Jr.’s 1954 book Theory of Algorithms, came from the Soviet Union. Markov’s book was translated into English, with US government sponsorship, by the Israel Program for Scientific Translations; the translation was published in 1961, although some Americans learned of Markov’s work prior to this time.1 This scientific exchange continued for decades. In 1979, Andrei Petrovich Ershov and Donald E. Knuth organized an international symposium that brought both Western and Soviet computer scientists on a “pilgrimage,” as Knuth called it, to the presumed homeland of the mathematician and astronomer Muḥammad ibn Mūsā al-Khwārizmī—the Khorezm region in Uzbekistan, then part of the USSR—to discuss “Algorithms in Modern Mathematics and Its Applications.”2 Participants debated a range of questions, such as how to define algorithm and whether algorithmic methods were fundamentally different from algebraic ones. The thinking on the two sides of the Iron Curtain showed different affinities—Soviet computer scientists were more influenced by Markov’s theory, which was probably not of decisive significance in the West—but the common ground was not insubstantial, and so the concept of algorithm cannot be said to be specific to Western countries or capitalist economies.

然而，算法理论确实反映了冷战时期资本主义制度中存在的紧张关系。在美国，一些最早的可编程计算机是由大学和政府研究机构开发的，尽管IBM很早就参与其中。随着20世纪50年代私营企业作用的增强，计算机研究人员对营利性企业对其领域的明显入侵表示不满。早期计算机科学的一些重要人物，尤其是在西欧，在某种程度上也在美国，是一些对纯数学比实际应用更感兴趣的学者。这种对实用性的贬低正符合学术界的价值等级制度。与17世纪的代数学家一样，早期的程序员面临着这样的指责：他们的工作仅仅是一项实用技能，他们必须证明其在大学学科中的地位。在工业界，程序员也面临着将他们的工作发展成一种工程形式的压力，这意味着他们要摒弃临时的解决方案和非正式分享的“民间传说”，转而采用基于科学原理的系统方法。

The theory of algorithms did, however, come to reflect tensions that existed in Cold War–era capitalist institutions. In the United States, some of the first programmable computers were developed at universities and government research facilities, although IBM had a stake from early on. As the role of private enterprise increased in the 1950s, computer researchers expressed resentment over the perceived intrusion of for-profit businesses into their field. Some major figures in early computer science, especially in Western Europe and to some extent in the United States, were academics more interested in pure mathematics than in practical applications. This devaluation of practicality fit the value hierarchies of academe. Much like algebraists in the seventeenth century, early programmers faced the accusation that their work was a mere practical skill, and they had to justify its place among the university disciplines. In industry, too, programmers faced pressure to develop their work into a form of engineering, which meant eschewing ad hoc solutions and informally shared “folklore” in favor of systematic methods based on scientific principles.

正如我在本章中所述，“算法”这一概念（后来被应用于计算机科学）通过在实际计算与数学和形式逻辑等高深理论之间架起一座桥梁，缓解了这些学科焦虑。由于人们认为他们的工作缺乏严谨性，一些早期程序员试图将计算过程形式化，以便能够用数学方法分析其正确性、效率和优雅性。这种方法遭到了其他人（通常在军事和商业机构而非大学工作）的反对，他们强调沟通和团队合作的重要性——而形式化方法并不适合处理这些编程方面的问题。最终，该学科达成了妥协。到了20世纪80年代，编程语言已经开始将算法逻辑与后来被称为“人为因素”的因素区分开来。通过人为因素，团队可以进行沟通，并使系统更容易被人们理解。这种区别有效地涵盖了形式主义兴起之前符号方法所伴随的政治问题，以类似于布尔所采用的术语，在计算机代码的假定逻辑方面和交流的社会复杂性之间建立了一条学科界限。

As I show in this chapter, the concept of algorithm, as the word came to be used in computer science, assuaged these disciplinary anxieties by creating a bridge between practical computation and the lofty theories of mathematics and formal logic. In response to the perception that their work lacked rigor, some early programmers sought to formalize computational procedures so that their correctness, efficiency, and elegance could be analyzed mathematically. This approach faced opposition from others (often working in military and commercial institutions rather than universities) who emphasized the importance of communication and teamwork—aspects of programming that formal approaches were ill suited to address. The discipline ultimately settled on a compromise. By the 1980s, programming languages had come to distinguish algorithmic logic from the “human factors,” as they came to be called, by which teams could communicate and systems be made comprehensible to people. This distinction effectively bracketed the political questions that had attended symbolic methods prior to the rise of formalism, establishing, in terms similar to those employed by Boole, a disciplinary line between the putatively logical aspects of computer code and the social complexities of communication.

在构建算法理论时，计算机科学家参考了计算机时代之前不久就已经开始的关于计算的讨论。20世纪30年代，阿隆佐·丘奇、埃米尔·波斯特和艾伦·图灵独立开发的计算（丘奇术语）、问题求解（波斯特术语）或计算（图灵术语）的数学模型。3丘奇的理论使用方程式定义数学函数，然后可以通过他称之为“算法”的规则进行变换。波斯特和图灵的模型基于对人类计算机所进行的心理过程的分析。虽然他们回答了纯数学问题，但这些理论包含了一些后来被编程语言采用的元素：丘奇贡献了一种新的函数符号，而图灵则展示了虚拟化的可能性，即一台机器可以模拟另一台机器。此外，丘奇和图灵还提供了一个用于数学推理算法属性的框架。

In assembling a theory of algorithms, computer scientists referred to a conversation about computation that had begun slightly before the computer era. In the 1930s, Alonzo Church, Emil Post, and Alan Turing had independently developed mathematical models of calculation (Church’s term), problem solving (Post’s), or computation (Turing’s).3 Church’s theory defines mathematical functions using equations, which may then be transformed by rules that he calls an “algorithm.” The models of Post and Turing are based on analyses of the mental processes undertaken by human computers. Although they responded to questions in pure mathematics, these theories contain elements that were later adopted by programming languages: Church contributed a new notation for functions, whereas Turing showed the possibility of virtualization, of a machine that could simulate another machine. Church and Turing provided, moreover, a framework for reasoning mathematically about the properties of algorithms.

然而，丘奇、波斯特和图灵的论文并没有对算法给出明确的定义；事实上，图灵的论文根本没有使用这个术语。4这个术语的第一个现代解释之一出现在丘奇的学生斯蒂芬·克莱恩于 1943 年发表的一篇论文中，他用算法理论来指代判断某类命题是真还是假的方法。5用克莱恩的话来说，这种“理论征服”已经应用于“ a 能被 b 整除”这样的命题，因为我们有可靠的方法对给定的a和b检验此类陈述。6马尔可夫在他的《算法理论》中进行了更深入的研究，给出了非正式和正式的定义。非正式地说，算法是具有三个特征的程序：确定性、通用性和结论性。7也就是说，它们必须定义精确，必须适用于一系列输入，并且必须在给定适当输入的情况下可靠地产生所需的输出。马尔可夫的书提出了一种算法的形式化理论，即用于转换字符序列的替换规则的集合——他反复使用无意义的例子“papagiglema”——从给定的字母表中抽取。

The papers of Church, Post, and Turing did not, however, provide explicit definitions of algorithm; indeed, Turing’s paper did not use the term at all.4 One of the first modern explications of the term appeared in a 1943 paper by Church’s student Stephen Kleene, who uses algorithmic theory to refer to methods for deciding whether propositions of certain types are true or false.5 This sort of “theoretical conquest,” as Kleene puts it, has been obtained for propositions of the form “a is divisible by b,” since we have a reliable means of testing such statements for a given a and b.6 Markov went into greater depth in his Theory of Algorithms, giving both informal and formal definitions. Informally, algorithms are procedures with three characteristics: definiteness, generality, and conclusiveness.7 That is, they must be precisely defined, must be applicable to a range of inputs, and must reliably produce the desired output given a proper input. Markov’s book presents a formal theory of algorithms as collections of substitution rules for transforming sequences of characters—he repeatedly uses the nonsensical example “papagiglema”—drawn from a given alphabet.

克莱尼和马尔可夫的定义不仅表明了算法概念的进一步扩展，也表明了优先顺序的转变。在十九世纪后期，“算法”主要被视为实用方法，或使实数和虚数等新型数学实体变得易于理解的方式。现在，核心问题是有效性——即通过精确指定且可在有限时间内完成的操作产生正确结果的属性。对有效程序的兴趣部分源于对布尔所称的发现真理的“指导性方法”的长期追求，但也反映了对此类项目局限性的新认识。1931年，库尔特·哥德尔证明，任何强大到足以表示算术的公理系统要么不一致，要么留下一些无法判定的问题。8正如马尔可夫指出，直到某些问题的程序不存在成为一个问题时，才需要严格的计算程序理论。9 在哥德尔之后，什么可以计算，什么不能计算成为一个紧迫的问题，计算规则的性质获得了新的理论意义。

The definitions of Kleene and Markov indicate not just a further expansion of the concept of algorithm but also a shift in priorities. Through the late nineteenth century, “algorithms” were primarily valued either as practical methods or as ways of making new types of mathematical entity such as real and imaginary numbers accessible. Now, the central issue was effectiveness—that is, the property of producing the correct result through operations that are precisely specified and that can be performed in a finite amount of time. The interest in effective procedures stemmed in part from a continuation of the long-standing quest for a “directive method,” as Boole had called it, for discovering truths, but it also reflected a newfound awareness of the limitations of such projects. In 1931, Kurt Gödel had shown that any axiomatic system powerful enough to represent arithmetic must either be inconsistent or leave some problems undecidable.8 As Markov points out, there was no need for a rigorous theory of computational procedures until the nonexistence of procedures for certain problems became an issue.9 After Gödel, what could be computed and what could not became a pressing question, and the nature of computational rules gained a newfound theoretical significance.

20 世纪 30 年代到 50 年代人们对算法的兴趣源于数学基础的哲学问题，这些问题最初与实际计算没有直接关系。但算法一词在当时也有更日常的含义。大约从 1900 年开始，欧几里得计算最大公约数的程序第一次被称为欧几里得算法，算法一词也适用于最小二乘等统计方法的计算。10 1905年的一本小册子介绍了一系列用于计算铁路线位置的“算法”（图 5.1）。这种用法并不典型；这类指令通常被称为计算计划。然而，称它们为算法并非概念上的重大飞跃。尽管从G.W.莱布尼茨到恩斯特·施罗德的数学家们都曾借用“算法”一词来指代类似代数的系统，但它的本意并没有被遗忘。中世纪的算法提供了解决数值问题的技术，而这些技术的扩展如今在工程、会计，尤其是蓬勃发展的统计学领域中都具有广泛的重要性。到1906年，统计学领域已经拥有了庞大的人工计算机团队，他们正在Brunsviga加法机上忙碌地工作。11

The interest in algorithms between the 1930s and the 1950s responded to philosophical issues about the foundations of mathematics, which initially had little direct bearing on practical computation. But the term algorithm also had a more workaday meaning at the time. Starting around 1900, Euclid’s procedure for computing the greatest common divisor began, for the first time, to be called the Euclidean algorithm, and the word algorithm was also applied to the calculations for statistical methods such as least squares.10 A pamphlet from 1905 presents a series of “algorithms” for use in computing the locations of railway lines (figure 5.1). This usage was atypical; such instructions were more often called computing plans. Yet calling them algorithms would not have been a major conceptual leap. Even though mathematicians from G. W. Leibniz to Ernst Schröder had appropriated the word algorithm to refer to algebra-like systems, its original meaning was not forgotten. The medieval algorism had provided techniques for solving numerical problems, and extensions of these techniques were now of widespread importance in engineering, accounting, and especially the growing field of statistics, which by 1906 had large teams of human computers cranking away at Brunsviga adding machines.11

到了20世纪后期，算法的理论意义和实践意义发生了碰撞。在计算机科学中，算法既是可以用逻辑数学方法分析的抽象概念，也是解决问题的指令。这个概念的混合体与数学有着渊源上的联系，但它也正在演变成一种新的东西，而定义它则成了某种家庭手工业。计算机科学家唐纳德·E·克努斯（Donald E. Knuth）在其多卷本著作《计算机编程艺术》（第一部分出版于1968年）中给出了基于五个标准的定义，我将在后面详细讨论这些标准：有限性、确定性、输入、输出和有效性。12亚历山德拉·I·福赛斯（ Alexandra I. Forsythe）和她的同事在1969年的教科书中将算法比作菜谱和舞蹈编排，两者都将复杂的过程分解为简单的步骤。13三年后，哈罗德·S·斯通（Harold S. Stone）提出了另一个非正式的解释：“任何机器人可以遵循的指令序列。” 14在乌兹别克斯坦举行的研讨会上，苏联数学家弗拉基米尔·乌斯宾斯基和阿列克谢·谢苗诺夫对这个问题提出了截然不同的看法，他们认为算法具有其他数学实体所不具备的语义特征。15与方程式不同，算法采用祈使语句的形式，敦促任何愿意倾听的人：去做吧！比定义更重要的是，这些文本汇集了一种算法理论，它提供了推理过程的方法，而到本世纪末，这些过程将成为那个时代的主导技术之一。

By the later twentieth century, these theoretical and practical senses of algorithm had collided. Algorithms, as they are studied in computer science, are at once abstractions that may be analyzed by logicomathematical means and instructions for solving problems. This conceptual hybrid had a genealogical connection to mathematics, but it was also turning into something new, and defining it became something of a cottage industry. In his multivolume work The Art of Computer Programming, the first installment of which was published in 1968, the computer scientist Donald E. Knuth gives a definition based on five criteria, which I will discuss in detail later: finiteness, definiteness, input, output, and effectiveness.12 In their 1969 textbook, Alexandra I. Forsythe and her collaborators compare algorithms to recipes and dance choreographies, both of which break complex processes down into simple steps.13 Three years later, Harold S. Stone offered another informal gloss: “any sequence of instructions that can be obeyed by a robot.”14 At the symposium in Uzbekistan, the Soviet mathematicians Vladimir Uspensky and Alexei Semenov presented a very different take on the issue, arguing that algorithms have a semantic character absent in other mathematical entities.15 Unlike equations, that is, algorithms take the form of imperative statements, urging whoever will listen: do this! More important than definitions, these texts assembled a theory of algorithms that provided methods of reasoning about the processes that by the end of the century would become one of the dominant technologies of an age.

本章阐述了算法的概念是如何在计算机编程话语中得到体现的。在考虑了 20 世纪早期的相关发展之后，我将重点介绍早期的编程语言 ALGOL，它的名字通常被解释为“算法语言”。从 1958 年 ALGOL 推出开始，程序员们就采用“算法”一词来指代介于民间知识和数学实体之间的某种东西：可以被多人重复用于多种用途的计算过程。起初，这个词被视为“程序”的同义词，但到了 20 世纪 80 年代，它开始指代一些更空灵的东西，指代与其实现和概念解释无关的抽象过程。这种抽象依赖于学科界限：一方面是算法系统的“硬逻辑”，另一方面是使这些系统在社会情境中运作的“软”关注点。我希望表明，这种划分是 ALGOL 及其众多后继语言（包括 Python、C++ 以及当今常用的众多其他编程语言）设计的基础，并且它有助于算法轻松地从一个应用环境迁移到另一个应用环境。在全面解释这一点之前，我必须先讨论一下早期计算机科学所借鉴的逻辑和数学思想的一些方面。

This chapter offers an account of how the idea of algorithm came to be enshrined in the discourse of computer programming. After considering relevant developments in the early twentieth century, I focus on the early programming language ALGOL, whose name is usually explained as algorithmic language. Starting around the time of ALGOL’s introduction in 1958, programmers adopted the term algorithm to refer to something midway between folk knowledge and mathematical entity: computational procedures that could be reused by multiple people and for multiple purposes. At first, the word was treated as a synonym of program, but by the 1980s it had come to refer to something more ethereal, to abstract procedures considered apart from their implementations and conceptual interpretations. This abstraction depends on a disciplinary boundary: on the one side is the “hard logic” of algorithmic systems, on the other the “soft” concerns of making those systems work in social contexts. This division, I hope to show, is fundamental to the design of ALGOL and its many successors, including Python, C++ and numerous other programming languages in common use today, and it contributes to the ease with which algorithms may be transferred from one application context to another. Before I can explain this point fully, I must discuss some aspects of the logical and mathematical ideas on which early computer science drew.

悖论与个人意志

Paradox and the Individual Will

关于逻辑形式主义遭遇的标准描述大致如下。伯特兰·罗素和阿尔弗雷德·诺斯·怀特黑德在《自然哲学的数学原理》中的巨大努力让读者感到不满意，而戴维·希尔伯特和威廉·阿克曼则试图做得更好。16他们的项目旨在产生一套完整一致的公理以及一个明确定义的程序（现在可以称之为算法，虽然略带时代错误）来判断给定的数学陈述是否成立。对希尔伯特项目的早期批评出现在 LEJ Brouwer 的著作中，他提倡一种他称之为直觉主义的反形式主义方法，他的工作对图灵产生了影响；这个学派拒绝排中律，因此不接受任何不能通过某种可实现的程序构造的数学实体的存在。17哥德尔、丘奇和图灵在短短几年内证明了希尔伯特纲领的关键要素是不可能的。在1936年发表的论文中，丘奇和图灵各自证明了希尔伯特的决策算法不可能存在：任何形式系统必然会留下一些无法解决的问题。悖论的出现带来了计算的第一个精确定义，这后来成为理论计算机科学的核心部分。

The standard account of what happened to logical formalism goes something like this. Bertrand Russell and Alfred North Whitehead’s monumental effort in Principia struck readers as unsatisfactory, and David Hilbert and Wilhelm Ackerman tried to do better.16 Their project was supposed to produce a complete and consistent set of axioms along with a clearly defined procedure (one can now say with only slight anachronism, an algorithm) for deciding whether given mathematical statements were true. An early critique of Hilbert’s project appeared in the work of L. E. J. Brouwer, who advocated for an antiformalist approach he called intuitionism, and whose work would be an influence on Turing; this school of thought rejects the law of excluded middle and thus does not accept the existence of any mathematical entity that cannot be constructed through some realizable procedure.17 Gödel, Church, and Turing, within the span of just a few years, proved key elements of the Hilbertian program impossible. In papers published in 1936, Church and Turing proved independently that Hilbert’s decision algorithm cannot exist: any formal system must leave some problems unsolvable. The advent of paradox produced, as a side effect, the first precise definitions of computation, which would later form a central part of theoretical computer science.

在这场思想骚动中，又一次出现了用符号取代文字的乌托邦式尝试。哥德尔在维也纳大学的导师之一是鲁道夫·卡尔纳普，他于 1928 年出版的《世界的逻辑结构》（通常简称为Aufbau，取自其德文名称Der logische Aufbau der Welt），概述了一种将复杂概念简化为简单概念的“构造体系”。18卡尔纳普将这个项目比作莱布尼茨的普遍特征，旨在通过将所有科学置于相同的概念基础上来统一它们。19虽然这个项目对算法技术理论的重要性不如丘奇和图灵的工作，但它值得首先考虑，因为它展示了莱布尼茨的程序在二十世纪初发展迅速。遵循罗素和怀特海的做法，卡尔纳普用方程式定义符号：

Caught up in this intellectual tumult was yet another utopian attempt to replace words with symbols. One of Gödel’s mentors at the University of Vienna was Rudolf Carnap, whose 1928 book The Logical Structure of the World, usually referred to as the Aufbau (after its German title, Der logische Aufbau der Welt), outlines a “constructional system” in which complex concepts are reduced to simpler ones.18 This project, which Carnap likens to Leibniz’s universal characteristic, is supposed to unify the sciences by placing them all on the same conceptual basis.19 While this project is not as significant to the technical theory of algorithms as the work of Church and Turing, it is worth considering first because it shows how the Leibnizian program was faring in the early twentieth century. Following the practice of Russell and Whitehead, Carnap defines symbols using equations:

114.品质之间的相似性（Sim）

构造： Sim = _Df α̂β̂ {α, β ∈ qual. α ↑ β ⊂ Ps}

释义：如果两个质量类别中的每个元素都与另一个质量类别中的每个元素部分相似，则称这两个质量类别相似。20

114. Similarity Between Qualities (Sim)

Construction: Sim =_Df α̂β̂{α, β ∈ qual. α ↑ β ⊂ Ps}

Paraphrase: Two quality classes are called similar if each element of one of them is part similar to each element of the other.20

该方程旨在构建一个关于性质相似性的概念，它展示了如何将使用这一概念的陈述转化为仅包含先前定义的概念（即性质和部分相似性）的陈述。卡尔纳普只针对他认为非常基础的概念创建了这样的公式；当他深入到物理现实和“文化对象”时，他仅仅给出了一个大概的描述。

The equation is supposed to construct a concept of similarity of qualities by showing how statements using this concept can be transformed into ones that involve only previously defined concepts—namely, quality and part similarity. Carnap managed to create such formulae only for what he took to be very basic concepts; when he advances to physical reality and “cultural objects,” he gives only a sketch.

至少就其目标而言，《构造》最接近于复兴孔多塞侯爵的乌托邦计划。与侯爵一样，卡尔纳普希望通过摒弃既定教条、支持科学理性来实现社会平等。卡尔纳普的早期工作包括“概念工程”——试图建立一个更好的概念体系，他将此视为百科全书启蒙计划的延续。21哲学家理查德·杰弗里后来将卡尔纳普描述为一位唯意志论者；在卡尔纳普看来，语言是一种概念技术，应该根据其对特定目的的效用来判断，并在可能时进行改进。22与侯爵非常相似，卡尔纳普认为历史是朝着技术发展和政治平等进步的故事，这种进步将体现在科学的通用语言中。

At least in its aims, the Aufbau was as close as anyone came to reviving the Marquis de Condorcet’s utopian project. Like the marquis, Carnap wanted to enact social leveling by expelling received dogmas in favor of scientific rationality. Carnap’s early work consisted of “conceptual engineering”—an attempt to build a better system of concepts which he saw as a continuation of the Enlightenment project of the Encyclopédie.21 The philosopher Richard Jeffrey later described Carnap as a voluntarist; for Carnap’s purposes, language was a conceptual technology that ought to be judged by its utility for particular purposes and improved when possible.22 Much like the marquis, Carnap regarded history as a story of progress toward both technological development and political equality, progress that would be embodied in a scientific universal language.

然而，卡尔纳普并没有扭转19世纪逻辑范围的缩小。在《构造》开篇，他指出他将从“传统视角”——即在他介入之前人们讨论科学的既有语言——来讨论一些问题。23他没有赋予这种视角任何认识论意义；他后来指出，“纯粹传统的”概念“客观上只是偶然的（就像一个国家的历史边界一样）。24然而，为了使新的语言更容易被人们理解，我们必须与这些历史遗产相联系。这种对传统的让步使卡尔纳普与孔多塞区别开来，后者试图避免使用任何先前的惯例。尽管卡尔纳普公开敌视伊曼纽尔·康德，但《构造》与“传统”的协商，暴露了康德《纯粹理性批判》之后改变逻辑学的根本区别的影响：个体获得知识并拥有知识本身。“尽管所有知识的主观起源都在于经验的内容及其联系，”卡尔纳普写道，但人们仍然可以“推进到一个主体间性的客观世界，这个世界可以从概念上被理解，并且对所有观察者而言都是相同的。” 25对卡尔纳普而言，科学无需完全摆脱既定的思维方式即可具有普遍性；相反，主体与客体的划分提供了一个安全的空间来容纳文化的影响。

Yet Carnap did not reverse the narrowing of the scope of logic that occurred in the nineteenth century. Near the opening of the Aufbau, he notes that he will discuss some issues from the “traditional viewpoint,” meaning the preexisting language in which people discussed science before his intervention.23 He assigns no epistemological significance to this viewpoint; “merely traditional” concepts, he later states, are “objectively speaking, merely accidental (just like the historical boundaries of a state).”24 Yet it is necessary to engage with these historical legacies to make the new language comprehensible to people. This concession to tradition differentiates Carnap from Condorcet, who wanted to avoid making any use whatsoever of prior conventions. Although Carnap was overtly hostile toward Immanuel Kant, the Aufbau’s negotiation with “tradition” betrays the influence of the fundamental distinction that transformed logic in the aftermath of Kant’s Critique of Pure Reason: a distinction between the way an individual arrives at knowledge and the knowledge itself. “Even though the subjective origin of all knowledge lies in the contents of experiences and their connections,” Carnap writes, one can still “advance to an intersubjective, objective world, which can be conceptually comprehended and which is identical for all observers.”25 Science, for Carnap, need not achieve total autonomy from received ways of thinking to be universal; instead, the subject–object divide provides a safe place to stow the influence of culture.

哥德尔的证明迫使卡尔纳普重整旗鼓，他后来的努力与构造的方向不同。他在 1934 年出版的《语言的逻辑句法》一书中第二次重要尝试发明一种通用语言。书名的改变意义重大：卡尔纳普不再声称要描绘世界的结构。相反，他进一步走向了唯意志论，坚持认为“我们在语言形式方面在各方面都拥有完全的自由；句子的构造形式和转换规则……都可以相当任意地选择。” 26这种“自由”使得人们能够对含义尚不清楚的形式进行实验：“让任何公设和推理规则都任意选择，那么这种选择，无论是什么，都将决定赋予基本逻辑符号什么含义。” 27卡尔纳普将此视为一个新想法，尽管它与乔治·皮科克一个世纪前在代数中采用的方法并非完全不同。区别在于卡尔纳普在多大程度上允许规则的选择是任意的，而不是受制于正确性或真理的普遍规律。无论费迪南德·德·索绪尔如何看待自然语言，形式语言的运作规则完全取决于个人。

Gödel’s proof forced Carnap to regroup, and his later efforts took a different tack from that of the Aufbau. He made a second major attempt at a universal language in his 1934 book The Logical Syntax of Language. The title change is significant: Carnap is no longer claiming to limn the structure of the world. Instead, he has moved further into voluntarism, maintaining “that we have in every respect complete liberty with regard to the forms of language; that both the forms of construction for sentences and the rules for transformation . . . may be chosen quite arbitrarily.”26 This “liberty” enables experimentation with forms whose meanings are not yet known: “let any postulates and rules of inference be chosen arbitrarily, then this choice, whatever it may be, will determine what meaning is to be assigned to the fundamental logical symbols.”27 Carnap presents this as a new idea, although it is not wholly dissimilar to the approach George Peacock had taken in algebra a century before. The difference is in the extent to which Carnap permitted the choice of rules to be arbitrary rather than governed by universal laws of correctness or truth. Whatever Ferdinand de Saussure may say about natural languages, formal languages work by rules that are wholly up to the individual.

在《逻辑句法》中，卡尔纳普通过区分对象语言和句法语言，设定了这种唯意志论的界限。28对象语言是正在构建的新符号系统；句法语言（也称为元语言）是用来描述它的语言——即德语或英语，并辅以一些特殊符号。这种区分的迹象可以从孔多塞侯爵的言论中找到：有时有必要用文字来解释符号的含义；此外，人们还可以注意到，这位侯爵实际上也在使用法语来描述他的体系。但卡尔纳普的研究对象是主客体划分，而他的前辈并没有这种划分。孔多塞的通用语言被认为是所有人都能理解的，因为它建立在自然理性的基础上；因此，对他来说，人们如何理解这些符号，既是系统本身的结构问题，也是认识论问题。相比之下，卡尔纳普更倾向于将句法语言视为务实地说：符号最初如何解释仅仅是一个主观问题，与对象语言的客观性无关，后者源于其受正式规则的支配。

In Logical Syntax, Carnap set the bounds of this voluntarism by means of a distinction between the object language and the syntax language.28 The object language is the new symbolic system being constructed; the syntax language (also called the metalanguage) is the language used to describe it—that is, German or English, supplemented with a few special symbols. A hint of this division may be found in the Marquis de Condorcet’s remark that it would sometimes be necessary to use words to explain the meanings of symbols; the marquis was also, one might further note, actually using French to describe his system. But Carnap was working within a subject–object division that was not in place for his precursor. Condorcet’s universal language was supposed to be comprehensible to all because of its grounding in natural reason; how people came to understand the symbols was thus, for him, an epistemological matter just as much as the structure of the system itself. Carnap, by contrast, treated the syntax language more pragmatically: how the symbols are initially explained is a merely subjective matter irrelevant to the objectivity of the object language, which stems, instead, from its governance by formal rules.

卡尔纳普的乌托邦计划虽然与计算机科学有着间接的联系，但却未能实现。卡尔纳普与阿尔弗雷德·塔斯基等同时代人一起，提出了一个在 ALGOL 项目中发挥核心作用的想法，即形式语言的想法，用塔斯基的话来说，这种语言“每个表达式的意义都由其形式唯一决定”。29然而，卡尔纳普更关心的是精确地表示经验知识，而不是计算，在这方面，他的系统在目的上与编程语言不同。30图灵以及波斯特、丘奇和克莱恩的工作在形式语言理论和计算理论之间架起了一座桥梁。虽然图灵对卡尔纳普的“构造”等总体系统的僵化感到恼火，但他的计算模型仍然保留着与卡尔纳普将其系统的“客观”方面与历史情境中的人类思想区分开来的主客体二元论相同的思想。正如卡尔纳普对“传统观点”做出让步一样，图灵也为直觉留下了空间，从而使机械过程能够与人类思维协同工作。

Carnap’s utopian plans would not be realized, although they do have an oblique relation to computer science. Along with contemporaries such as Alfred Tarski, Carnap contributed to an idea that would play a central role in the ALGOL project—the idea of a formal language in which, as Tarski put it, “the sense of every expression is uniquely determined by its form.”29 Carnap was, however, concerned less with computation than with precisely representing empirical knowledge, and in this regard his systems are different in purpose from programming languages.30 Turing’s work, along with that of Post, Church, and Kleene, provided a bridge between the theory of formal languages and computation. Although Turing chafed at the rigidity of totalizing systems like Carnap’s Aufbau, his model of computation inhabits much the same subject–object dualism by which Carnap divided the “objective” aspects of his system from historically situated human thought. Just as Carnap made concessions to the “traditional viewpoint,” Turing left room for intuition, thus enabling mechanical processes to work together with human thought.

图灵的计算模型基于一个后来被称为图灵机的虚构装置。这台机器由一个读写头组成，它沿着一条被分成离散方块的纸带左右移动，并在移动过程中读取、擦除和写入符号。在每一点，机器都处于有限个状态之一；它的行为由一个表格决定，该表格指定了机器在每个状态下的操作，具体操作取决于它从纸带上当前方块中读取的符号。机器的状态被认为代表了执行计算的人的“心理状态”；图灵认为有限个状态就足够了，因为如果存在无限多种可能性，那么其中一些就必须被“混淆”。31他将纸带描述为“纸张的类似物”，这意味着，就像数学家的笔记本一样，它提供了一个记录解决方案的地方，以及“用于‘辅助记忆’的粗略笔记” 。32利用这个概念模型，图灵证明了两件看似矛盾的事情。一是存在一种“通用计算机器”——一种可以通过从磁带上读取其表格编码来模拟任何其他图灵机的图灵机。33另一是存在一些问题，这些问题无法通过任何足够精确的规则来解决，以至于这种机器无法执行这些问题。

Turing’s model of computation is based on an imaginary device that later came to be known as the Turing Machine. This machine consists of a read–write head that moves left and right along a paper tape divided into discrete squares, reading, erasing, and writing symbols as it goes. At each point, the machine is in one of a finite number of states; its behavior is determined by a table that specifies what it will do in each state, based on which symbol it reads from its current square on the tape. The state of the machine is supposed to represent the “state of mind” of a person performing a computation; Turing argues that a finite number of states is sufficient because if there were infinitely many possibilities, some of them would have to be “confused.”31 He describes the tape as “the analogue of paper,” meaning that, like the notebook of a mathematician, it provides a place to record the solution along with “rough notes to ‘assist the memory.’”32 Using this conceptual model, Turing proved two seemingly contrary things. One is the existence of a “universal computing machine”—a Turing Machine that can simulate any other Turing Machine by reading an encoding of its table from the tape.33 The other is the existence of problems that cannot be solved by any rules precise enough to be performed by such a machine.

一些流行的说法不谨慎地​​将图灵模型描述为“通用机器”，暗示它可以完成任何事情，但对于图灵来说，该模型更多的是一种设定学科界限——作为关于什么算作计算、什么不算作计算的陈述。图灵对计算的特殊定义方式使其与原始算法紧密相关。首先，图灵机采用的“书写”形式涉及离散符号序列，而不是任何形式的图形表示——而不是（比如说）相对长度代表数值关系的线条。在这方面，图灵的理论与印地语-阿拉伯数字系统一致，而与欧几里得著作中创立的古典几何传统背道而驰。此外，图灵机的符号词汇量有限——机器不可能在运行过程中创造新的符号。图灵认为这种限制是合理的，因为基本符号可以组合成无数个“复合符号”，他将这种做法与数字和字母书写联系起来：“像 17 或 999999999999999 这样的阿拉伯数字通常被视为单个符号。同样，在任何欧洲语言中，单词都被视为单个符号（然而，中文试图拥有可枚举的无限个符号）。” 34正如图灵所说，我们引入的符号越多，其中两个符号看起来相似并因此混淆的可能性就越大；因此，通过少量基本符号表达数据可以最大限度地提高清晰度。

Some popular accounts have incautiously described Turing’s model as a “universal machine,” suggesting that it could accomplish anything and everything, but for Turing the model functions more as a way of setting disciplinary lines—as a statement about what counts as computation and what does not. Turing’s particular way of defining computation aligns it strongly with the original algorism. To begin with, the form of “writing” employed by Turing Machines involves sequences of discrete symbols, not any sort of pictorial representation—not (say) lines whose relative lengths represent numerical relations. In this regard, Turing’s theory is aligned with the Hindi–Arabic numeral system and against the classical geometric tradition founded in the work of Euclid. Moreover, a Turing Machine is limited to a fixed, finite vocabulary of symbols—there is no possibility for the machine to coin, as it were, a new symbol in the course of its operation. Turing justifies this limitation on the grounds that the basic symbols may be combined into infinitely many “compound symbols,” a practice that he links to numerals and alphabetical writing: “an Arabic numeral such as 17 or 999999999999999 is normally treated as a single symbol. Similarly in any European language words are treated as single symbols (Chinese, however, attempts to have an enumerable infinity of symbols).”34 The more distinct symbols we introduce, as Turing has it, the greater the chance that two of them will look alike and thus become confused; expressing data through a small number of basic symbols thus maximizes clarity.

图灵对书写的看法还有更深层的特质。当机器“读取”纸带上的一个方块时，它只需识别出写在那里的符号类型，并遵循相应的规则：如果符号是“A”，就这样做；如果符号是“D”，就那样做。符号的形状只在彼此区分时才重要；它们特定的视觉特征——例如数字“9”类似于倒置的数字“6”——并不重要，任何可能存在于其使用中的文化传统也同样重要，除非这些传统以某种方式反映在数据或规则的设计中。因此，人们可以很容易地构建一台违反既定语义实践的图灵机——比如，可以创建一台加法机，交换数字“1”和“2”，这样“19 + 1 = 32”。这样的系统可能违反直觉，以至于实际上毫无用处，但这种考虑在图灵的计算理论中是没有的。用一个词来说，图灵机在使用符号时是符合常规的：没有任何东西，甚至社会协议，对哪些符号可以在系统中扮演哪些角色施加任何限制。

Turing’s view of writing has a further idiosyncrasy. When the machine “reads” a square of the tape, all it has to do is identify the type of symbol written there and follow the corresponding rule: if the symbol is “A,” do this, if the symbol is “D,” do that. The shapes of the symbols only matter in their distinguishability from each other; their specific visual qualities—the fact that 9 resembles an inverted 6—are of no relevance, nor are any cultural traditions that may exist regarding their use, except insofar as those traditions are in some way reflected either in the data or in the design of the rules. As a result, one can easily set up a Turing Machine that violates established semantic practices—one can create an adding machine, say, that swaps the digits 1 and 2, so that 19 + 1 = 32. Such a system might be so counterintuitive as to be effectively useless, but such considerations have no place in Turing’s theory of computation. In their use of symbols, Turing Machines are, to coin a word, aconventional: nothing, not even social agreement, places any restriction on which symbols may play which roles in the system.

然而，这种非传统性必然有其限度。显然，计算实践的某些方面（无论是人类的还是机械的）都受到符号的传统含义的影响：人们通常不会选择构建交换1和2的系统，即使理论上可以。主客体划分的影响正是在此显现出来。关于图灵虚构机器的编程，与《逻辑句法》中的卡尔纳普一样，图灵是一位唯意志论者：他将计算规则视为个人选择的对象，选择依据是它们如何适用于特定目的。图灵在 1938 年发表的博士论文中最明确地阐述了这一观点。在论文中，他思考了对哥德尔不完备定理的潜在回应。给定一个逻辑系统L，可以构造另一个逻辑系统L ₁并添加一些额外元素；然后将相同过程应用于L ₁以生成L _{2 ，依此类推。通过判断要沿着这个阶梯向上攀登多远，可以解决至少一些无法在原始系统}L中解决的问题。

This aconventionality must, however, have limits. Clearly, some aspects of computational practices (be they human or mechanical) are influenced by the conventional meanings of symbols: people do not usually choose to construct systems that swap 1 and 2, even if they theoretically could. It is here that the influence of the subject–object divide becomes apparent. With respect to the programming of his imaginary machine, Turing was, like the Carnap of Logical Syntax, a voluntarist: he viewed computational rules as objects of individual choice, to be selected based on how well they suit a particular purpose. Turing states this view most explicitly in his doctoral thesis, published in 1938. In the thesis, he considers a potential response to Gödel’s incompleteness theorem. Given a logic system L, one constructs another logic system L₁ that adds some additional element; one then applies the same procedure to L₁ to produce L₂, and so forth. By exerting judgment about how far up this ladder to ascend, one can solve at least some problems that cannot be addressed within the original system L.

这一论点本质上是通过放弃形式主义来应对不完备性的。正如图灵所解释的那样，他的目的是“选择一个特定的建设性逻辑系统用于实际用途”；因此，将他的方法形式化“无异于本末倒置”。35这一论点与图灵的导师路德维希·维特根斯坦的观点相呼应，维特根斯坦后来认为规则永远不可能自给自足——人们总是需要更多规则来遵循规则。36图灵甚至考虑了一些不可能遵循的规则，例如假设一个图灵机可以访问“神谕”，可以立即回答某些（可能无法计算的）问题。37哪些计算系统可用且有用，无法通过应用更多形式规则来解决，这只会增加抽象级别而无法解决任何问题。 （很容易通过反证证明，如果一个问题的可判定性本身是不可判定的，我们永远无法找到答案。）如同后期维特根斯坦一样，图灵通过放弃对绝对基础的探索，转而将系统置于其社会背景中，解决了这一倒退。最终，对图灵而言，计算规则回应了使用者的人类目的，而这些目的要么服务于这些目的，要么不服务于这些目的。

This argument responds to incompleteness, in essence, by backing away from formalism. As Turing explains, his purpose is “choosing a particular constructive system of logic for practical use”; as a result, formalizing his method “would be putting the cart before the horse.”35 This argument resonates with the work of Turing’s mentor Ludwig Wittgenstein, who would later argue that rules can never be self-sufficient—one always would need more rules for how to follow the rules.36 Turing even considers some rules that would be impossible to follow, such as a hypothetical Turing Machine with access to an “oracle” that can instantly answer certain (possibly uncomputable) questions.37 Which computational systems are usable and useful cannot be settled by applying more formal rules, which would only increase the level of abstraction without resolving anything. (It is easy to prove by contradiction that if the decidability of a problem is itself undecidable, we can never find that out.) Like late-period Wittgenstein, Turing resolves this regress by abandoning the search for absolute foundations and instead situating systems within their social contexts. Ultimately, for Turing, computational rules answer to the human purposes of the people who use them, which they either serve or do not.

虽然一些后来的评论家推测图灵将人类思维视为一种生物图灵机，但该论文表明，至少在 1938 年，他将其视为一种不太可靠的预言机——一种补充了无法简化为机械规则的附加元素的图灵机。38他写道，确定数学命题的真假需要“两种能力的结合，我们可以称之为直觉和独创性”。39直觉涉及“做出自发的判断，这不是有意识的推理的结果”，而独创性涉及“通过适当安排命题，或许还有几何图形或图纸来辅助直觉”。40根据图灵的说法，形式逻辑的目的是消除由于直觉因人而异而引入的“任意性”。41他继续说，在“前哥德尔时代”，人们认为形式主义甚至可以完全消除直觉，但现在看来这是不可能的。42他的替代策略是接受直觉的作用，并发展“证明中并非所有步骤都是机械的，有些步骤是直觉的”实践。43

While some later commentators have presumed that Turing saw the human mind as a sort of biological Turing Machine, the thesis suggests that, at least in 1938, he saw it more as a somewhat unreliable oracle machine—as a Turing Machine supplemented by an additional element that cannot be reduced to mechanical rules.38 Determining the truth or falsity of mathematical propositions, he writes, involves “a combination of two faculties, which we may call intuition and ingenuity.”39 Intuition involves “making spontaneous judgments which are not the result of conscious trains of reasoning,” whereas ingenuity involves “aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.”40 The purpose of formal logic, according to Turing, is to remove the “arbitrariness” introduced by the fact that intuition varies from person to person.41 In “pre-Gödel times,” he continues, it was believed that formalism could be taken so far as to eliminate intuition altogether, but this is now seen to be impossible.42 His alternative tactic is to embrace the role of intuition and develop practices in which “not all the steps in a proof are mechanical, some being intuitive.”43

图灵这方面的工作对计算机科学的学科实践产生了被低估的影响，正如我们将看到的，计算机科学在算法的合理性方面赋予了所谓的“直觉”或“常识”以重要的地位。然而，图灵的计算模型只是大约在同一时期出现的几个模型之一，而新兴的计算理论远不止他对印地语-阿拉伯语算法的推广。相反，丘奇和他的学生克莱尼的贡献基于递归函数的概念——即一个函数以自身为单位进行定义，最终可以分解为单个值。一个简单的例子是以下阶乘的定义：

This aspect of Turing’s work would have an underrated influence on the disciplinary practice of computer science, which, as we will see, grants an important role to something called “intuition” or “common sense” in justifying algorithms. Turing’s model of computation was, however, only one of several to appear around the same time, and there was more to the emerging theory of computation than his generalization of the Hindi–Arabic algorism. The contributions of Church and his student Kleene were based, instead, on the idea of a recursive function—that is, a function defined in terms of itself in a way that is ultimately resolvable into a single value. A simple example would be the following definition of a factorial:

丘奇 1936 年的论文引入了一种称为λ演算（或 lambda 演算）的符号，它提供了一种在公式中当场构造函数的方法：不用单独的方程f ( x ) = M来定义f，而是可以使用λ x [ M ] 来引用该函数而不给出其名称。现在，这种符号的变体用于在包括 Python 在内的许多编程语言中构造“匿名”函数。图灵在 1936 年丘奇的文章正在审查中时发现了丘奇的工作，并添加了一个附录，证明这两个模型是等价的。44尽管存在这种形式上的等价性，但这些模型基于不同类型的符号方法。如果图灵通过描述排列数字的逐步说明来遵循原始算法，那么丘奇的演算则与代数更相似，涉及包含表示未指定值的字母的符号方程的转换。

Church’s 1936 paper introduces a notation called the λ-calculus (or lambda calculus), which provides a way of constructing functions on the spot within a formula: instead of defining f with the separate equation f (x) = M, one could use λx[M] to refer to this function without giving it a name. Variants of this notation are now used to construct “anonymous” functions in a number of programming languages, including Python. Turing found out about Church’s work while his 1936 article was under review and added an appendix proving that the two models are equivalent.44 In spite of this formal equivalence, the models are based on different types of symbolic method. If Turing follows the original algorism by describing step-by-step instructions for arranging digits, Church’s calculus has more kinship with algebra, involving the transformation of symbolic equations that include letters representing unspecified values.

值得注意的是，在 1936 年的论文中使用“算法”一词的是丘奇，而不是图灵。丘奇对这个术语的使用并不完全符合现代意义，因此值得研究。λ符号通常被视为一种形式化算法的方法，即将算法表示为递归函数，但丘奇并没有以这种方式呈现他的演算。“很明显，”他写道，“对于任何正整数的递归函数，都存在一种算法，使用该算法可以有效地计算出该函数的任何所需特定值”；具体而言，该算法包括执行一个归约过程，直到找到所需的结果。45如果我们把递归函数定义看作计算机程序，那么这个“算法”并非程序本身——它是一种运行程序的方法。在这方面，丘奇的著作展现了“算法”一词与公式运算的持续关联。丘奇无疑了解该词的本义，但他也深入研究了施罗德的工作，对施罗德来说，“算法”是代数关系的系统。46丘奇并没有按照施罗德特有的意义使用这个词，但他的用法仍然可以追溯到莱布尼茨的代数谱系。

It is notable that Church, not Turing, is the one who used the word algorithm in his 1936 paper. Church’s use of this term is not quite in line with the modern sense, and so it is worth examining. The λ notation is often taken as a means of formalizing algorithms by representing them as recursive functions, but Church does not present his calculus this way. “It is clear,” he writes, “that for any recursive function of positive integers there exists an algorithm using which any required particular value of the function can be effectively calculated”; specifically, the algorithm consists of performing a reduction procedure until the desired result is found.45 If we think of recursive function definitions as computer programs, then this “algorithm” is not the program itself—it is a method for running programs. In this regard, Church’s work exhibits the continuing association of the word algorithm with the manipulation of formulae. While Church was undoubtedly aware of the original sense of the word, he was also deeply engaged with the work of Schröder, for whom “algorithms” were systems of algebraic relations.46 Church does not use the word in Schröder’s idiosyncratic sense, but his usage can still be placed in the algebraic lineage going back to Leibniz.

人们必须根据其中一种模型来思考计算机器，这一点并非不言而喻。巴贝奇在丘奇和图灵之前一个世纪就已经设想出了可编程计算机，而一些早期计算机的设计者，例如康拉德·楚泽的 Z1（1936-38 年）和 IBM Mark I（1939-43 年），最初并不知道他们的工作。47约翰·冯·诺依曼通常被认为是标准计算机体系结构的设计者，他与图灵私交甚好，并且非常熟悉他关于决策问题的论文，但尚不清楚图灵的虚构机器是否对他的计划产生了任何重大影响。48然而，丘奇和图灵最终确实成为了计算机科学学科的共同参考点。他们提供的最重要的东西与其说是实际机器设计的范例，不如说是数学推理后来被称为“算法”的理论框架。

That one must think of computing machines in terms of either of these models is not self-evident. Babbage had already imagined a programmable computer a century before Church and Turing, and the designers of some early computers, such as Konrad Zuse’s Z1 (1936–38) and the IBM Mark I (1939–43), were initially unaware of their work.47 John Von Neumann, often held up as the designer of the standard computer architecture, knew Turing personally and was deeply familiar with his paper on the decision problem, but it is not clear that Turing’s imaginary machines had any strong influence on his plan.48 Yet Church and Turing did eventually become common reference points for the discipline of computer science. The most important thing they provided was less a paradigm for the design of actual machines than a theoretical framework for reasoning mathematically about what came to be called “algorithms.”

这门新学科直到 1954 年图灵去世后很久才牢固确立。计算历史学家写道，一场“软件危机”始于 20 世纪 60 年代，原因是人们认为程序变得过于复杂而不可信任。其中一个触发事件是 1962 年水手 1 号航天器的故障，显然是由于编码错误造成的。49 1968 年和 1969 年，北约召开了“软件工程”会议，与会者提出了防止此类错误并使大型系统更易于管理的方法。50正是在这场太空时代的骚动中，“算法”一词成为了一个重要的学科术语。在丘奇和图灵时代，“算法”的概念更多地与纯数学而非计算机器联系在一起，它有望通过将程序置于严格的数学分析之下，将编程从一项实用技能转变为一门科学。这种转变与有史以来最具影响力的形式主义系统之一——编程语言的发展密切相关。

This new discipline would not be firmly established until well after Turing’s death in 1954. Historians of computation write of a “software crisis” brought on, starting in the 1960s, by a perception that programs were growing too complex to be trusted. One triggering event was the failure of the Mariner 1 spacecraft in 1962, apparently due to a coding mistake.49 In 1968 and ’69, NATO held conferences on “software engineering” at which participants proposed ways of preventing such errors and making large systems more manageable.50 It was amid this space-age ferment that the word algorithm became a major disciplinary term. The idea of algorithm, which in the time of Church and Turing had been associated more with pure mathematics than with computing machines, promised to transform programming from a practical skill into a science by subjecting procedures to rigorous mathematical analysis. This transformation was closely tied to the development of one of the most influential systems ever to claim the mantle of formalism: the programming language.

一种算法语言

An Algorithmic Language

第一批可编程计算机是通过穿孔卡、纸带或直接操纵电气连接来控制的。在现代在计算机上，大多数编程都是用 C++、JavaScript 和 Python 等抽象语言完成的，这些语言不仅用于控制机器，还用于以人类可读的形式呈现计算过程。将编程语言不加批判地定义为我在本书中讨论过的符号方法的继承者是错误的；正如 Mark Priestley 所强调的，计算机代码与数学证明中使用的符号有不同的用途。51一个区别是赋值运算符，在某些编程语言中，它指示计算机更改变量的值——这种做法在现代代数中是不存在的。52沃伦·萨克 ( Warren Sack) 主张另一种谱系，即编程语言源自启蒙思想家用来编纂机械艺术知识的“工作语言” 。53但代数符号确实在编程语言的设计中发挥了重要作用，计算机代码提出了一些与通用代数方案中相同的符号学问题。编程语言必须将技术严谨性与交流清晰度结合起来，而它们划分这些功能的特殊方式借鉴了布尔时代首次出现的主客体二元论。

The first programmable computers were controlled through punch cards, paper tape, or direct manipulation of electrical connections. In modern computers, most programming is done in abstract languages such as C++, JavaScript, and Python that are designed not just for directing machines but also for presenting computational processes in a human-readable form. It would be a mistake to frame programming languages too uncritically as a successor of the symbolic methods I have discussed in this book; computer code, as Mark Priestley has emphasized, serves a different purpose from the notation used in mathematical proofs.51 One difference is the assignment operators that, in some programming languages, instruct the computer to change the value of a variable—a practice that is absent in modern algebra.52 Warren Sack has argued for an alternative genealogy in which programming languages descend from the “work languages” by which Enlightenment thinkers codified knowledge of the mechanical arts.53 But algebraic symbols did play a major role in the design of programming languages, and computer code raises some of the same semiotic questions that arose in universal algebra schemes. Programming languages must combine technical rigor with communicational clarity, and the particular way they came to divvy up these functions draws on the subject–object dualism that first emerged in the time of Boole.

虽然编程语言总是将计算和交流结合在一起，但两者之间的界限最初是可以改变的。早期的程序员通过流程图等符号来表示计算过程，这些符号起源于赫尔曼·H·戈德斯坦和约翰·冯·诺依曼在 20 世纪 40 年代末的工作。54然而，这些符号如果不先转换成数字代码就无法输入计算机。格蕾丝·霍珀在她 1952 年的文章《计算机教育》中指出，这种翻译的需求已经成为一种乏味的干扰；数学家没有专注于手头的问题，而是“变成了程序员”。55因此，霍珀提议创建我们现在所说的编译器——一种自动将人类可读的指令序列转换成机器可执行形式的程序。她写道，随着这种翻译器的引入，“程序员可能会重新成为数学家”。56正如普里斯特利、诺弗雷和阿尔伯特所表明的那样，在电子计算的早期阶段，术语是不同的。57霍珀只在提及机器代码时使用“程序”一词；她将人类可读的形式称为“计算机信息”。用她的话来说，“程序员”是指代理人——目前通常指女性——其任务是将数学转化为机器代码。一旦这个过程自动化，程序员就变成了机器，而不是人类。58

While programming languages always combined computation and communication, the boundary between the two was initially malleable. Very early programmers represented computational processes through notations such as flowcharts, which originated in the work of Herman H. Goldstine and John Von Neumann in the late 1940s.54 Yet these notations could not be fed into a computer without first being translated into numerical codes. In her 1952 article “The Education of a Computer,” Grace Hopper argues that the need for this translation has become a tedious distraction; rather than focusing on the problem at hand, the mathematician “has become a programmer.”55 Hopper thus proposes the creation of what we now call a compiler—a program that automatically translates a human-readable sequence of instructions into a machine-executable form. With the introduction of this translator, she writes, “the programmer may return to being a mathematician.”56 As Priestley, Nofre, and Alberts have shown, the terminology was different in this early phase of electronic computation.57 Hopper only uses the word program when referring to machine code; her term for the human-readable form is “computer information.” The “programmer,” in her language, is the agent—typically, at this point, a woman—whose task is to translate mathematics into machine code. Once this process is automated, the machine, not the human, is the programmer.58

霍珀的文章标志着人类迈向通过类似语言的方式控制机器的第一步。她总结道，她一直在研究的计算机——UNIVAC，目前拥有“霍珀希望，她能为学生提供与大学二年级学生完全同等水平的扎实数学教育”，并希望很快能将其提升到研究生水平。59计算机的能力确实会扩大，尽管并不总是朝着她预测的方向发展。从1955年开始，霍珀指导开发了两种编程语言，MATH-MATIC 和 FLOW-MATIC；后者演变为 COBOL，即“常见的商业导向语言” 。60尽管她在《计算机教育》中重点关注数学，但 FLOW-MATIC 和 COBOL 代码中却很少包含数学符号，而是采用诸如“ ADD a TO b”之类的英语祈使句形式。正如 COBOL 这个名字所表明的那样，这种语言主要用于商业用途，其设计反映了五角大楼等机构官僚的沟通惯例。

Hopper’s article marked an early step toward enabling people to control machines through something resembling language. She concludes that the computer she had been working on, the UNIVAC, currently has “a well grounded mathematical education fully equivalent to that of a college sophomore” and hopes that it will soon advance to the graduate level.59 The capacities of computers would indeed expand, although not always in the direction she predicted. Starting in 1955, Hopper directed the development of two programming languages, MATH-MATIC and FLOW-MATIC; the latter evolved into COBOL, which stood for “COmmon Business Oriented Language.”60 In spite of her focus on mathematics in “The Education of a Computer,” FLOW-MATIC and COBOL code contains little mathematical notation, instead taking the form of English imperative sentences such as “ADD a TO b.” As the name COBOL indicates, the language was primarily intended for business uses, and its design reflected the communicational conventions of bureaucrats at institutions such as the Pentagon.

正如 COBOL 的例子所示，编程语言与数学符号并无必然联系。代码也可以借鉴英语。然而，数学符号却成为了编程语言的主要来源。1954 年，由约翰·巴克斯 (John Backus) 领导的 IBM 团队起草了编程语言 FORTRAN 的第一个规范，FORTRAN 是“IBM 数学公式翻译系统”的缩写；该语言于 1957 年投入商业使用。61 FORTRAN 的出现部分是为了回应这样一个问题：“机器能否以足够低的成本将足够丰富的数学语言翻译成足够经济的程序，从而使整个过程可行？” 62对经济性的考虑至关重要。一些计算机操作员对“自动编程”系统持怀疑态度，因为至少在非常早期的编译器中，自动生成的机器代码的准备和执行速度通常都比手写代码慢。尽管存在这些担忧，COBOL 和 FORTRAN 等编译型语言在人类可理解性方面仍具有巨大优势，这促使它们得到了广泛的采用。而且，标准化的编程语言（至少在理论上）可以使相同的指令用于不同型号的计算机，从而使代码更易于共享和重用。

As the example of COBOL shows, programming languages have no necessary connection to mathematical symbols. Code could just as well draw on the English language. Mathematical notation did, however, become a major source for programming languages. In 1954, an IBM group led by John Backus drafted the first specification of the programming language FORTRAN, which stood for “The IBM Mathematical FORmula TRANslating system”; the language became commercially available in 1957.61 FORTRAN arose in part in response to the question: “Can a machine translate a sufficiently rich mathematical language into a sufficiently economical program at a sufficiently low cost to make the whole affair feasible?”62 The concern with economy is crucial. Some computer operators were skeptical of “automatic programming” systems because, at least with very early compilers, automatically generated machine code was often slower both to prepare and to execute than handwritten code. Despite these concerns, compiled languages such as COBOL and FORTRAN had major advantages in regard to human intelligibility that encouraged their widespread adoption. What is more, a standardized programming language would (at least in theory) enable the same instructions to be used with different models of computer, making code easier to share and reuse.

这一标准化成为另一种略有不同的编程语言 ALGOL 的主要关注点。与大约在同一时期出现的 LISP（LISP 处理器）一样，ALGOL主要由学者而非商业或军事机构的员工开发，它反映了将数学的严谨性和形式逻辑应用于计算机编程的愿望。第一个版本 ALGOL 58 是由一个委员会设计的，该委员会于 1957 年在苏黎世举行的美国计算机协会 (ACM) 和德国应用数学与力学学会 (GAMM) 联合会议上成立。63 ALGOL经历了后来有两个修订版，即 ALGOL 60 和 ALGOL 68。ALGOL 这个名字通常被解释为算法语言，尽管它最初的名字是国际代数语言；1958 年的一项提案将其描述为遵循“数学公式符号”惯例的“描述计算过程的通用语言”。64 ALGOL不仅应该像 FORTRAN 一样充当可编译语言，而且还应该充当在1958 年 1 月创刊的 ACM 通讯杂志上发布计算过程的符号。虽然 ALGOL 从未像 COBOL 和 FORTRAN 那样被广泛采用，但它影响了许多仍然流行的编程语言，包括 C、C++ 及其许多衍生语言。

This standardization became a primary focus of another, somewhat different programming language: ALGOL. Like LISP (LISt Processor), which appeared around the same time, ALGOL was developed largely by academics rather than employees of business or military institutions, and it reflected a desire to apply the rigor of mathematics and formal logic to computer programming. The first version, ALGOL 58, was designed by a committee organized at the 1957 joint meeting of the US-based Association for Computing Machinery (ACM) and the German Society of Applied Mathematics and Mechanics (GAMM) in Zürich.63 ALGOL went through two later revisions known as ALGOL 60 and ALGOL 68. The name ALGOL is usually explained as algorithmic language, although the original name was International Algebraic Language; a 1958 proposal described it as a “universal language for the description of computing processes” that followed the conventions of “mathematical formula notation.”64 ALGOL was supposed to serve not just as a compilable language like FORTRAN but also as a notation for publishing computational procedures in the journal Communications of the ACM, which launched in January 1958. While never as widely adopted as COBOL and FORTRAN, ALGOL influenced numerous programming languages that remain popular, including C, C++, and their many derivatives.

算法语言(德语为algorithmische Sprache )这一短语的采用可能受到 ALGOL 委员会成员 Heinz Rutishauser 的影响。Rutishauser 之前曾在 ERMETH 计算机方面工作过，这是瑞士联邦理工学院从 20 世纪 40 年代末开始开发的。在 1955 年的一篇期刊文章中，他描述了一种“算法写作”的形式，ERMETH 的用户可以通过这种形式指定他们想要完成的计算。65借鉴康拉德·楚泽(Konrad Zuse) 的工作，楚泽开发了一种名为Plankalkül的复杂计算计划符号，Rutishauser 的系统包含许多将成为 ALGOL 元素的思想，例如for循环和if-then语句。66这类语言是否应该被称为“算法”是有争议的；巴克斯更喜欢“代数”一词。67然而， “算法语言”这个短语最终指的是一种特定的计算机语言风格——一种基于分支、循环指令序列的语言，与早期版本的 COBOL 不同，其书写方式类似于数学符号。

The adoption of the phrase algorithmic language (in German, algorithmische Sprache) was probably due to influence of the ALGOL committee member Heinz Rutishauser. Rutishauser had previously worked on the ERMETH, a computer developed by the Swiss Federal Institute of Technology starting in the late 1940s. In a 1955 journal article, he described a form of “algorithmic writing” by which users of the ERMETH could specify the computations they wanted done.65 Drawing on the work of Konrad Zuse, who had developed a sophisticated notation for computing plans called Plankalkül, Rutishauser’s system contained a number of ideas that would become elements of ALGOL, such as for loops and if–then statements.66 That such languages should be called “algorithmic” was controversial; Backus preferred the term “algebraic.”67 Yet the phrase algorithmic language eventually came to refer to a certain style of computer language—one based on branching, looping sequences of instructions, written (in contrast to early versions of COBOL) in something resembling mathematical notation.

ALGOL 被ACM 通讯社正式采用，我认为这是确保算法被广泛采用为计算过程的通用术语的主要因素。68早期的ACM出版物包含许多过程，但它们使用各种术语来指代，包括过程、方法或子例程，并且通常以散文和符号表示法的组合来描述。在 1960 年 2 月的《CACM》杂志中，该期刊增加了一个名为“算法”的新部分，最初由美国国家标准局的 Joseph Henry Wegstein 编辑。该部分征求读者分享程序和过程，以及其有效性的“证明”和批评性反馈；要符合收录条件，过程必须用 ALGOL 编写，并附有非正式注释，说明它们的作用和使用方法。在开源许可证的前身中，该部分包含允许免费重复使用的版权声明。大多数过程都涉及数值这些问题或许会引起数学家的兴趣，尽管并非所有问题都与数字有关。著名的快速排序算法正是在这里首次发表的，该算法后来成为了教科书的主流。69

ALGOL’s official adoption by Communications of the ACM was, I would venture, a major factor in securing the widespread adoption of algorithm as a general term for computational procedures.68 Early ACM publications included numerous procedures, but they are referred to by various terms, including procedure, method, or subroutine, and often described in a combination of prose and symbolic notation. In the February 1960 issue of CACM, the journal added a new section called “Algorithms,” initially edited by Joseph Henry Wegstein of the US National Bureau of Standards. The section solicited readers to share programs and procedures as well as “certifications” of their effectiveness and critical feedback; to be eligible for inclusion, procedures had to be written in ALGOL and accompanied by an informal comment indicating what they do and how to use them. In a precursor of open-source licenses, the section included a copyright notice permitting free reuse. Most of the procedures addressed numerical problems that would have interested mathematicians, although not all of them specifically dealt with numbers. It was in this venue that the famous Quicksort algorithm, which would later become a textbook mainstay, was first published.69

正如编者按所解释的，本节的目标是创建一个程序员可以利用的标准程序“库”。70这种标准程序的积累——算法概念中一个被低估的方面——将成为软件工程的核心原则。在1968年的北约会议上，马尔科姆·道格拉斯·麦克罗伊 (Malcolm Douglas McIlroy) 呼吁“大规模生产软件组件”，通过提供可在各种情况下重复使用的共享模块集合，使开发更加系统化。71尽管CACM集合没有完全达到 McIlroy 的标准，但它符合这种普遍的思维方式，鼓励程序员设计可在初始应用程序之外重新利用的程序。值得注意的是，虽然这些“算法”是用 ALGOL 语言发布的，但它们不仅用于基于 ALGOL 的系统——读者报告说他们用其他语言（例如 FORTRAN）测试了这些程序。72创建这个开放论坛鼓励人们将算法视为通用解决方案的共同库，可以在各种编程语言、各种硬件和各种用途中重复使用。它还鼓励人们反思什么是好的算法，因为需要一些编辑标准；因此，该部分促进了算法理论的发展。

The goal of this section, as an editorial note explained, was to create a “library” of standard procedures that programmers could draw on.70 This accumulation of standard procedures—an underappreciated aspect of the concept of algorithm—would become a central principle of software engineering. At the 1968 NATO conference, Malcolm Douglas McIlroy called for “mass produced software components,” which would make development more systematic by providing a shared collection of modules that could be reused in various contexts.71 Although the CACM collection did not fully live up to McIlroy’s standards, it fit this general way of thinking by encouraging programmers to design procedures that could be repurposed beyond their initial applications. It is notable that, while they were published in ALGOL, the “algorithms” were not only used with ALGOL-based systems—readers reported testing the procedures in other languages, such as FORTRAN.72 Creating this open forum encouraged people to view algorithms as a common stock of general solutions that could be reused in various programming languages, with various hardware, and for various purposes. It also encouraged reflection about what made for a good algorithm, since some editorial standards were needed; the section thus spurred on the development of a theory of algorithms.

ALGOL 的兴起恰逢编程在知识层面日益提升的地位。在计算机时代早期，硬件被视为工程师的专长，而编程则被贬低为乏味乏味的学科，常常被归类为女性化的秘书工作。73如今，学术界开始认真关注 ALGOL，他们必须证明将其纳入大学学科的合理性。在后来的一篇文章中，尼克劳斯·沃思 (Niklaus Wirth) 讲述了 1963 年加州大学伯克利分校的氛围，当时他正在研究 ALGOL。他写道，沃思的研究小组面临着阻力，因为“有传言说，其中一些人既不懂欧姆定律，也不懂麦克斯韦方程组”——也就是说，他们不了解这些机器设计所依据的电气原理。74 沃思用温迪·全惠京 (Wendy Hui Kyong Chun) 详细讨论过的一种修辞手法，将这种情况描述为一种巫术。75 “从远处看，那个编译器简直令人恐惧；兴奋恰恰来自于对一种无人能完全理解的机制的洞察，”沃思回忆道。“它散发着一种独特的魔法气息。” 76这种情况与 19 世纪中叶代数学家面临的情况截然相反，当时批评家们担心，随着符号方法的采用，这门科学正在变成一种机械的实践而非智力。如今，计算的物理基础濒临消失在抽象的云雾之中。

The rise of ALGOL coincided with the increasing intellectual status of programming. Early in the computer era, hardware was viewed as a subject fit for engineers, whereas programming was denigrated as tedious and often coded as a feminine, secretarial practice.73 Now, academics were turning serious attention to it, and they had to justify its inclusion among the university disciplines. In a later article, Niklaus Wirth recounts the atmosphere of the University of California, Berkeley in 1963, where he worked on ALGOL. Wirth’s research group, he writes, faced resistance because “word passed that some of these people did neither know Ohm’s law nor Maxwell’s equations”—that is, they did not understand the electrical principles on which the machines were designed.74 Using a rhetoric that Wendy Hui Kyong Chun has discussed in detail, Wirth describes this situation as involving a kind of wizardry.75 “Looking at it from the distance, that compiler was a horror; the excitement came precisely from having insight into a machinery that nobody understood fully,” Wirth recalled. “There was the distinct air of sorcery.”76 This situation is the converse of the one algebraists faced in the mid-nineteenth century, when critics worried that, with the adoption of symbolic methods, the science was becoming a mechanical practice rather than an intellectual one. Now, the physical basis of computation threatened to vanish in a cloud of abstraction.

尽管最初面临阻力，计算机科学家最终还是克服了对抽象出机械结构的顾虑；Wirth 后来宣称：“编程的本质就是抽象。” 77 Wirth 回忆说，伯克利的 ALGOL 程序员的趋势是“发现算法的基本概念，将它们从不同语言特性的各种体现中提取出来，并以纯粹、精炼的形式呈现，摆脱任意和限制性的适用规则。” 78 ALGOL 有时被描述为一种面向问题的语言，而不是面向计算机的语言。79它的目的是使计算机操作员能够将注意力从电路转移到算法上——即不依赖于任何特定机器性质的通用程序。根据 Wirth 的说法，这种趋势的一个例子是 Adriaan van Wijngaarden 对假设的广义 ALGOL 的描述，其目标（非常符合查尔斯·巴贝奇的精神）是“通用中的简单性” 。80

In spite of this initial pushback, computer scientists eventually did overcome their reservations about abstracting out machinery; “the essence of programming,” Wirth would later declare, “is abstraction.”77 The trend among ALGOL programmers at Berkeley, Wirth recalls, “was to discover the fundamental concepts of algorithms, to extract them from their various incarnations in different language features, and to present them in a pure, distilled form, free from arbitrary and restrictive rules of applicability.”78 ALGOL was sometimes described as a problem-oriented language as opposed to a computer-oriented language.79 Its purpose was to enable computer operators to turn their attention from circuitry to algorithms—that is, general procedures that do not depend on the nature of any particular machine. An example of this trend, according to Wirth, was Adriaan van Wijngaarden’s description of the hypothetical Generalized ALGOL, which aimed (very much in the spirit of Charles Babbage) at “simplicity in generality.”80

广义 ALGOL 从未实现，但 ALGOL 的三个主要版本制定了一种让巴贝奇都感到自豪的泛化形式。它的基本元素对于了解常见现代编程语言的读者来说会很熟悉，尽管它们之间存在一些细微的差别。81我将在这里重点介绍 ALGOL 60，因为它是最流行的版本。ALGOL 60 程序通常使用变量，这些变量用于存储可以随时间变化的数据（与代数变量不同）。变量具有必须声明的类型，例如“整数 A ”。可以使用赋值运算符 := 设置值，例如“ A := 5”。可以在赋值语句的右侧使用代数表达式，但不能在左侧使用。除其他外，该语言还包括仅在特定条件下执行操作的if - then语句和在数字序列中迭代的for循环。例如，“ for i := 2 step 1 until A do ” 会将i依次设置为从 2 到A （我们之前将其设置为 5）的每个值，增量为 1，并执行do之后的操作。我们可以将上述代码组合成以下完整程序：

Generalized ALGOL was never implemented, but the three main versions of ALGOL enact a form of generalization that would make Babbage proud. Its basic elements will be familiar to readers who know common modern programming languages, although there are some subtle differences.81 I will focus here on ALGOL 60 because it was the most popular version. ALGOL 60 programs usually employ variables, which are used to store data that (unlike algebraic variables) can change over time. The variables have types that must be declared, as in “integer A.” The values can be set using the assignment operator :=, as in “A := 5.” One can employ algebraic expressions on the right side of an assignment, although not on the left. The language also includes, among other things, if–then statements that perform operations only on certain conditions and for loops, which iterate through sequences of numbers. For instance, “for i := 2 step 1 until A do” will sequentially set i to each value from 2 to A (which we previously set to 5) by increments of 1, and perform the operation following do for each. We might assemble the above into the following complete program:

这将计算 5 的阶乘——即从 1 到 5 的所有整数的乘积——然后将其值打印到屏幕上。82

This will compute the factorial of 5—that is, the product of all the integers from 1 to 5—and then print its value to the screen.82

虽然这个程序只能计算一个特定数字（例如 5）的阶乘，但 ALGOL 代码通常被组织成“过程”，旨在尽可能广泛地应用。例如，这里有一个来自“算法语言 Algol 60 修订报告”的过程：83

While this program can only compute the factorial of one specific number (e.g., 5), ALGOL code is usually organized into “procedures” that are intended to be applicable as widely as possible. Here, for instance, is a procedure from the “Revised Report on the Algorithmic Language Algol 60”:83

• 过程 Absmax ( a ) 大小：( n , m ) 结果：( y ) 下标：( i , k ) ；
• 注释 将大小为 n 乘以 m 的矩阵 a 的绝对最大元素转移到 y，并将该元素的下标转移到 i 和 k；
• 数组 a；整数 n，m，i，k；实数 y；
• 开始整数 p , q；
• y：= 0；
• 对于 p := 1步骤1直到 n 执行对于 q := 1步骤1直到 m 执行
• 如果 abs ( a [ p , q ]) > y 则开始 y := abs ( a [ p , q ]) ; i := p ; k := q 结束 结束 Absmax

• procedure Absmax (a) size: (n,m) Result: (y) Subscripts: (i,k) ;
• comment The absolute greatest element of the matrix a, of size n by m is transferred to y, and the subscripts of this element to i and k ;
• array a ; integer n, m, i, k ; real y;
• begin integer p, q ;
• y := 0 ;
• for p := 1 step 1 until n do for q := 1 step 1 until m do
• if abs(a[p, q]) > y then begin y := abs(a[p, q]) ; i := p; k := q end end Absmax

此过程是“高水位线”算法的一种变体，该算法通过逐个遍历列表中的数字并记录迄今为止最大的数字来查找列表中的最大值。该过程适用于一系列输入，从而（根据马尔可夫的非正式定义）成为一种算法，其原因在于它是一种抽象：待处理的矩阵未指定，而仅称为。这种做法是弗朗索瓦·韦达 (François Viète) 于 1591 年提出的分析方法的巅峰——使用任意字母来指代未指定的值，以便呈现具有完全普遍性的结果。

This procedure is a variant of the “high water mark” algorithm, the procedure of finding the largest number in a list by going through the numbers one by one and keeping a record of the largest one yet. What makes the procedure applicable to a range of inputs and thus (by Markov’s informal definition) an algorithm is a certain kind of abstraction: the matrix to be processed is not specified but rather referred to only as a. This practice is the culmination of the analytical method François Viète introduced in 1591—the use of arbitrary letters to refer to values that are left unspecified so as to present a result in its full generality.

在解释这个系统时，ALGOL 的创建者借鉴了 20 世纪初已经成型的形式语言的思想。ALGOL 最具影响力的创新之一是用于描述其语法的形式化符号，后来被称为巴科斯-诺尔范式。例如，if - then语句定义如下：84

In explaining this system, the creators of ALGOL drew on the idea of formal language that had solidified in the early twentieth century. One of ALGOL’s most influential innovations was the formal notation used to describe its syntax, which later came to be known as the Backus–Naur form. For instance, the if–then statement is defined as follows:84

〈if 子句〉 ::= if〈布尔表达式〉then

〈无条件语句〉 ::= 〈基本语句〉 | 〈复合语句〉 | 〈块〉

〈if 语句〉 ::= 〈if 子句〉 〈无条件语句〉

〈if clause〉 ::= if 〈Boolean expression〉 then

〈unconditional statement〉 ::= 〈basic statement〉 | 〈compound statement〉 | 〈block〉

〈if statement〉 ::= 〈if clause〉 〈unconditional statement〉

第一行指示如何通过在if和then之间放置布尔表达式来构造“if 子句” ；第二行提供了用于构造“无条件语句”的三个选项列表（以“|”分隔）；第三部分将两者结合成“if 语句”的定义。修订报告中关于语法的部分强调了准确性，其引言来自维特根斯坦的《逻辑哲学论》：“能说的都可以说清楚，不能说的就必须保持沉默。” 85巴克斯借鉴塔斯基等逻辑学家的研究成果，还勾勒出了一种形式语义学（从未完全完成），用于指定 ALGOL 程序的计算“含义”——即它们告诉机器做什么。86

The first line indicates how an “if clause” may be constructed by placing a Boolean expression between an if and a then; the second provides a list of three options (separated by “|”) for constructing an “unconditional statement”; the third combines these two into a definition of “if statement.” Indicating an emphasis on precision, the section on syntax in the Revised Report takes its epigraph from Wittgenstein’s Tractatus Logio-Philosophicus: “What can be said at all can be said clearly, and whereof one cannot speak thereof one must be silent.”85 Drawing on the work of logicians such as Tarski, Backus also sketched out a formal semantics (never fully completed) that would specify the computational “meanings” of ALGOL programs—that is, what they tell the machine to do.86

这种形式化方法有一个值得强调的特点。与 FORTRAN 和 COBOL 一样，ALGOL 允许在标识符中使用单词——变量和其他程序元素的名称，例如上例中的过程名Absmax。然而，这些单词的含义在计算语义中不起作用。修订报告的作者列出了五个示例标识符，这些标识符似乎是为了强调其完全的任意性：“q”、“Soup”、“V17a”、“a34kTMNs”和“MARILYN”。87他们解释说：“标识符没有固有含义，但用于识别简单变量、数组、标签、开关和过程。它们可以自由选择”，只要它们不干扰内置功能即可。88示例代码主要遵循代数符号的约定，使用单个字母作为变量名，但出于显而易见的原因，它也用英文单词（例如“size”和“Result”）来注释这些字母。89因此，代码包含文字，但其计算“含义”与这些文字可能传达给人们的内容完全不同。

This approach to formalization has a peculiarity worth emphasizing. Like FORTRAN and COBOL, ALGOL allows for the use of words in identifiers—the names of variables and other program elements, such as the procedure name Absmax in the foregoing example. Yet the meanings of these words play no role in the computational semantics. The authors of the Revised Report list five example identifiers that seem calculated to emphasize their total arbitrariness: “q,” “Soup,” “V17a,” “a34kTMNs,” and “MARILYN.”87 “Identifiers,” they explain, “have no inherent meaning, but serve for the identification of simple variables, arrays, labels, switches, and procedures. They may be chosen freely,” provided that they do not interfere with built-in features.88 The sample code mostly follows the conventions of algebraic notation by using single letters for variable names, but it also annotates those letters with English words such as “size” and “Result,” for obvious reasons.89 The code, then, contains words, but its computational “meaning” is wholly separate from whatever those words might convey to people.

由于其标准化的野心，ALGOL 有时被描述为计算领域的“世界语”，一种描述算法的通用语言。90然而，更仔细的比较应该是约翰·威尔金斯的真正性格——一套书面符号系统，旨在表达人人共享的普遍思想，因此可以用任何口语朗读。ALGOL 58 和 60 有三种不同的“表示形式”：标准“参考”形式、设计用于输入机器的纯文本形式，以及允许使用希腊字母等精细符号的出版形式。前述示例为参考形式。正如修订报告所解释的那样，这三种形式“仅在符号的选择上”有所不同，而“结构和内容”相同。91同样，标识符可以随意替换而不会影响这种结构。因此，与真正的性格一样，ALGOL 寻求多样性中的统一性，表示所有人共有的思想，无论使用何种特定的词语来表达它们。

On account of its standardizing ambitions, ALGOL was sometimes described as an “Esperanto” of computing, a universal language for describing algorithms.90 A closer comparison, however, would be John Wilkins’s real character—the system of written symbols that were supposed to express universal ideas that were shared by everyone and that could therefore be read aloud in any spoken language. ALGOL 58 and 60 existed in three distinct “representations”: a standard “reference” form, a plaintext form designed for entry into machines, and a publication form that allows for niceties like Greek letters. The foregoing example is in the reference form. As the Revised Report explains, the three forms differ “only in the choice of symbols,” whereas “structure and content” are the same.91 Identifiers, likewise, could be swapped out at pleasure without affecting this structure. Like the real character, then, ALGOL seeks unity in multiplicity, representing ideas that are common to all regardless of the particular words used to express them.

这种通用性方法的局限性在后来关于 ALGOL 的讨论中变得更加明显。该语言的最终版本，ALGOL 68 旨在基于不同的语言提供本地化版本——在英语版本中可以写成if ，在俄语版本中可以写成если ( jésli ) 。92在某些情况下，英语 ALGOL 最终被用于使用其他语言的环境中；因此，在 Van Wijngaarden 的 ALGOL 68 代码中，我们发现了英语和荷兰语的混合。请考虑以下来自解谜程序的几行代码，它构造一个数组来跟踪当前正在使用的位置（gebruikt ），然后使用系统时钟记录当前时间（tijd ）： 93

The limitations of this approach to universality became more apparent in later discussions of ALGOL. The final version of the language, ALGOL 68, was intended to exist in localized variants based on different languages—one could write if in the English version and если (jésli) in the Russian version.92 In some cases, English ALGOL ended up being used in contexts where other languages were spoken; one thus finds in Van Wijngaarden’s ALGOL 68 code a mixture of English and Dutch. Consider these lines from a puzzle-solving program, which construct an array for keeping track of positions that are currently being used (gebruikt), then records the current time (tijd) using the system clock:93

这段斑驳的国际化历史表明，ALGOL 作为通用语言的地位，在很大程度上取决于这样一种理念：算法的存在独立于用来表达它们的特定词语或符号。像“.algol”这样的标识符GEBRUIKT可能对程序的语义没有影响，但词语仍然存在，也就不难理解为什么 Van Wijngaarden 更喜欢用他的母语输入它们。

This mottled history of internationalization shows the extent to which ALGOL’s status as a universal language depends on the idea that algorithms exist independently of the particular words or symbols used to express them. Identifiers like GEBRUIKT may make no difference to the semantics of the program, but the words are still there, and it is not hard to see why Van Wijngaarden preferred typing them in his native tongue.

创建一种“算法语言”来统治所有语言的想法不会实现。编程语言、方言和变体激增。后来的语言大多继续使用文字（通常是英语），尽管也有一些例外。1962 年，肯尼斯·E·艾弗森 (Kenneth E. Iverson) 发表了 APL（“一种编程语言”），它避开了文字，而倾向于使用←和 等符号；他后来解释说，他的目的是将布尔和弗洛里安·卡乔里等思想家在数学符号中发现的认知优势与编程语言的“通用性”结合起来。94但ALGOL 将形式语义置于表示细节之上，这一做法产生了持久的影响。有些人可能更喜欢 ALGOL 的外观a + b而不是 LISP (SUM,A,B)，但区别仅仅是 ALGOL 程序员 Peter Landin 所说的“语法糖”——帮助逻辑药物发挥作用的语言甜味剂。95其结果是，在认识论上占主导地位的形式观认为词语和符号完全是任意的，而在实践观中，它们发挥着不可否认的重要认知和交流功能，两者之间出现了断裂。在研究如何处理这种断裂的过程中，新兴的计算机科学学科被迫再次努力解决困扰数学家几个世纪的符号和语言问题。这一次，算法的概念变得至关重要。

The idea of creating one “algorithmic language” to rule them all would not come to fruition. Programming languages, dialects, and variants proliferated. Later languages mostly continued the use of words (often in English), although there were some exceptions. In 1962, Kenneth E. Iverson published APL (“A Programming Language”), which eschews words in favor of symbols such as ← and ; his purpose, as he later explained, was to combine the cognitive advantages that thinkers such as Boole and Florian Cajori had found in mathematical notation with the “universality” of programming languages.94 But ALGOL’s elevation of formal semantics over the details of presentation had a lasting influence. Some might favor the look of ALGOL’s a + b over LISP’s (SUM,A,B), but the difference is merely what the ALGOL programmer Peter Landin called “syntactic sugar”—linguistic sweetener to help the logical medicine go down.95 The result was a disconnect between an epistemologically privileged formal view in which words and symbols were wholly arbitrary and a practical view in which they played undeniably important cognitive and communicational functions. In working out how to manage this disconnect, the emerging discipline of computer science was forced to grapple, once again, with the questions about symbols and language that had bothered mathematicians for centuries. This time, the idea of algorithm became pivotal.

文化与开关电路

Culture and the Switching Circuit

那么，所有 ALGOL 变体所共有的算法“内容”究竟是什么呢？修订报告指出，ALGOL 60“适合以足够简洁的形式表达一大类数值处理过程，以便直接自动翻译成编程的自动计算机的语言” 。96然而，从一开始，编程语言的设计者就雄心勃勃，超越了单纯的数字运算。斯坦福大学计算机科学项目的创始人乔治·福赛斯 (George Forsythe) 在一次题为“计算机革命的教育意义”的讲座中强调，计算机不能局限于数学领域：“机器控制的二进制数字字符串可以模拟很多种事物，数字只是其中一种。” 97他继续说道，1 和 0 可以“模拟高速公路上的汽车、棋子、盒子里的电子、音符、俄语单词、纸上的图案、人体细胞、颜色、电路等等。” 98但一个领域不可能包罗万象；必须有一些区别于其他学科的东西。正如查德·韦尔蒙（Chad Wellmon）所指出的，现代学科的定义更多地取决于学者所参与的社群，而非其研究对象。99计算机科学（CS）学科共同体是围绕一系列实践形成的，这些实践虽然表面上可以应用于任何研究对象，但却与其他科学的实践截然不同。

What, exactly, is the algorithmic “content” that is supposed to be common to all the variants of ALGOL? The Revised Report states that ALGOL 60 is “suitable for expressing a large class of numerical processes in a form sufficiently concise for direct automatic translation into the language of programmed automatic computers.”96 Yet the designers of programming languages had, from the very beginning, ambitions beyond mere number crunching. The founder of Stanford’s computer science program, George Forsythe, stressed in a lecture on “Educational Implications of the Computer Revolution” that computers could not be contained within the field of mathematics: “Machine-held strings of binary digits can simulate a great many kinds of things, of which numbers are just one kind.”97 The ones and zeroes may, he continues, “simulate automobiles on a freeway, chess pieces, electrons in a box, musical notes, Russian words, patterns on a paper, human cells, colors, electrical circuits, and so on.”98 But a field cannot do everything; there had to be something distinguishing it from others. As Chad Wellmon has pointed out, modern disciplines are defined less by their objects of study than by the communities in which scholars participate.99 The disciplinary community of computer science (CS) was formed around a set of practices that, while they could apparently be applied to any object of study, were distinct from those of other sciences.

这个新学科的核心要素 (尽管其核心程度取决于问的对象) 是算法理论。1963 年，Juris Hartmanis 和 Richard E. Stearns (当时是通用电气的员工，尽管两人后来都成为了学者) 提出了计算复杂性理论的基础。100他们提出这个理论作为图灵工作的延伸，不仅考虑函数是否可计算，还考虑计算它们的难度。他们通过图灵机对给定大小的输入必须执行多少次操作来衡量这种难度 — — 一种抽象的时间概念，与计算硬件的具体情况无关。例如，假设我们正在对一个包含n 个项目的列表进行排序。最明显的排序算法平均需要与n2^成比例的操作数，因此对于长列表它们会变得非常慢。只要稍加巧思，我们就能设计出遵循较慢增长曲线n log n的算法，从而能够在合理的时间内处理更大量的数据（图 5.2）。同样的思路也适用于内存使用。复杂性理论发展成为一个分类系统，用于判断算法的效率和问题的难度。它还提出了一系列关于复杂性类别之间关系的难题，从而为理论计算机科学设定了长期的研究议程。

A central element of this new discipline (although how central depended on whom one asked) was a theory of algorithms. In 1963, Juris Hartmanis and Richard E. Stearns—then General Electric employees, although both later became academics—laid out the rudiments of computational complexity theory.100 They present this theory as an extension of Turing’s work that considers not just whether functions are computable but also how hard it is to compute them. They measure this difficulty by how many operations must be performed by a Turing Machine for input of a given size—an abstract conception of time that does not depend on the specifics of computing hardware. Suppose, for instance, that we are sorting a list with n items. The most obvious sorting algorithms require, on average, a number of operations proportional to n², and they can thus become very slow for long lists. With some ingenuity, we can devise algorithms that follow the slower-growing curve n log n, and that can thus handle much larger amounts of data in a reasonable amount of time (figure 5.2). The same thinking can also be applied to memory usage. Complexity theory developed into a classification system that is used to judge both the efficiency of algorithms and the difficulty of problems. It also opened a range of difficult questions about the relationships between complexity classes, thus setting a long-standing research agenda for theoretical computer science.

丘奇和图灵的工作为思考这些问题提供了基础，也为抽象处理算法提供了理论基础。图灵证明他的假想机器等价于丘奇的λ-演算，这只是众多等价证明中的第一个，这些证明表明各种计算形式可以互换。克莱尼后来提出了后来被称为丘奇-图灵论题的命题，这个无法证明但得到广泛支持的命题是，所有有效的计算方法都等价于这两个模型（因此彼此等价）。101这个论题（尽管伴随着哲学上的困难）提供了一个理论基础，即认为算法能够保持其基本特性，而不管使用什么形式系统来表达它们以及使用什么物理过程来实现它们。

The work of Church and Turing provided a basis for thinking about such issues and a rationale for treating algorithms abstractly. Turing’s proof that his imaginary machines were equivalent to Church’s λ-calculus was only the first of a number of equivalence proofs showing the interchangeability of various forms of computation. Kleene later articulated what came to be known as the Church–Turing Thesis, the unprovable but widely supported proposition that all effective methods of computation are equivalent to these two models (and thus to each other).101 This thesis (although attended with philosophical difficulties) provided a rationale for viewing algorithms as capable of preserving their fundamental identity regardless of what formal system is used to express them and what physical process is used to enact them.

这种相互转换性适用于编程语言，至少前提是它们足够强大，能够模拟图灵机。在1976年麻省理工学院（MIT）的一篇著名备忘录《Lambda：终极命令》中，Guy Lewis Steele Jr. 和 Gerald Jay Sussman 展示了如何将一些常见的 ALGOL 技术应用于可以翻译成 Scheme，一种类似 LISP 的编程语言，其灵感来自 Church 的计算模型。他们告诉我们，这些翻译“抓住了构造语义的本质”。102该项目的一个前提是，即使以不同的风格重写算法，它们也能保留其根本的“本质”（尽管作者显然对哪种风格最好有自己的看法）。这份备忘录中有一段题词（可能是杜撰的）归于亚伯拉罕·林肯，暗示这些风格之间的选择取决于个人喜好：“喜欢这种东西的人会发现这也是他们喜欢的东西。” 103

This mutual convertibility applies to programming languages, at least provided that they are powerful enough to simulate a Turing Machine. In a well-known 1976 Massachusetts Institute of Technology (MIT) memo titled “Lambda: The Ultimate Imperative,” Guy Lewis Steele Jr. and Gerald Jay Sussman show how a number of common ALGOL techniques can be translated into Scheme, a LISP-like programming language inspired by Church’s model of computation. These translations, they tell us, “capture the essence of the semantics of the construct.”102 A premise of this project is that algorithms can retain their fundamental “essence” even when they are rewritten in disparate styles (although the authors clearly have an opinion about which style is best). The memo has an epigraph attributed (probably apocryphally) to Abraham Lincoln, suggesting that the choice between these styles is a matter of personal preference: “People who like this sort of thing will find this is the sort of thing they like.”103

但图灵等价性仅扩展到一种语言理论上可以计算的数学函数范围。在其他方面，编程语言并不完全相同。ALGOL 与结构化编程范式相关，而 LISP 则鼓励函数式风格。其他语言，例如 Simula 和 Prolog，则走上了完全不同的道路。这些范式融入了机构文化：许多欧洲大学都在使用 ALGOL，而麻省理工学院是 LISP 的领地，FORTRAN 和 COBOL 则统治着私营部门。语言和编码标准的激增引发了有时被称为“圣战”的事件——特定方法的拥护者之间的激烈争论，有时甚至会因为像制表符应该由多少个空格组成这样琐碎的问题而发生争执。104尽管这些争论被广泛嘲笑，但怨恨的深度表明，与主观性相关的问题——美学、清晰度、组织的直观性——对于编程实践仍然至关重要，即使它们被排除在形式语义学之外。学科不能完全消除人性。他们所能做的就是将某些问题置于界限之外，让它们由个人选择、制度文化或教学权宜之计来决定，而不是让它们受到严格的认识论规范的约束。

But Turing equivalence only extends to the range of mathematical functions a language can theoretically compute. In other ways, programming languages are not all the same. ALGOL is associated with the paradigm of structured programming, whereas LISP encourages a functional style. Other languages such as Simula and Prolog tread entirely different paths. These paradigms were integrated into institutional culture: ALGOL was used at many European universities, whereas MIT was LISP territory and FORTRAN and COBOL ruled the private sector. The proliferation of languages and coding standards sparked what are sometimes described as “holy wars”—bitter exchanges between partisans of particular approaches, sometimes over issues as trivial as how many spaces should make up a tab.104 Although these disputes were widely mocked, the depth of the rancor shows that matters affiliated with subjectivity—aesthetics, clarity, intuitiveness of organization—remained crucial to the practice of programming, even as they were excluded from formal semantics. Disciplines cannot eliminate the human altogether. What they can do is place certain questions outside of bounds, leaving them to individual choice, institutional culture, or pedagogical expediency rather than subjecting them to rigorous epistemic norms.

在划定这一界限的过程中，计算机科学家们又一次经历了一场由来已久的争论，即符号和日常语言之间应该存在多少连续性。1965年，Jean E. Sammet（后来成为ACM首位女主席）发表了题为“英语作为编程语言的应用”的演讲，呼吁建立一种自然语言界面，使人们能够“直接与计算机交流，而无需学习某种专门的中间语言”，从而使计算更加高效、易于使用。105 她的想法是，编程语言最终将包含整个英语语法，从而完成Hopper在其编译器中开启的自动化进程——让机器接管编程中繁琐的部分，以便人们能够专注于手头的问题。通过倡导自然语言编码，Sammet 延续了Hopper的以人类的目的和制度为中心，这鼓励人们以交流为主要关注点的编程语言观点。

In setting up this boundary, computer scientists underwent yet another iteration of the ages-old debate about how much continuity should exist between symbols and ordinary language. In 1965, Jean E. Sammet (who later became the first woman president of the ACM) gave a talk entitled “The Use of English as a Programming Language,” in which she called for a natural-language interface that would make computing more efficient and accessible by enabling people to “communicate directly with the computer without having to learn some specialized intermediate language.”105 The idea was that programming languages would eventually incorporate the whole of English grammar, thus completing the process of automation that Hopper had begun with her compiler—letting the machine take over the tedious parts of programming so that people can focus on the problems at hand. By advocating for natural language coding, Sammet continued Hopper’s centering of human purposes and institutions, which encouraged a view of programming languages primarily focused on communication.

这种观点与巴克斯在 ALGOL 项目中采用的正式方法格格不入，并且遭到了那些拥护图灵式计算非传统性的人士的抵制。萨姆特的演讲记录了芬兰计算机科学家雷诺·库尔基-苏奥尼奥 (Reino Kurki-Suonio) 对此做出的略带戏谑的回应：

This view sat uneasily with the formal approach Backus had taken in the ALGOL project, and it faced pushback from those who embraced the aconventionality of Turing-style computation. The transcript of Sammet’s presentation records the following, somewhat tongue-in-cheek, response by the Finnish computer scientist Reino Kurki-Suonio:

我发现在严格定义的编程目的中使用英语单词和短语很方便，主要原因是英语不是我的母语（笑）。英语提供了大量的符号，可以赋予它们任何含义（更多笑声），而无需参考我以前的知识。106

The main reason I find it convenient to use English words and phrases for strictly defined programming purposes is that English is not my native language (laughter). English provides a huge store of symbols to which any meaning can be assigned (more laughter) without referring to my former knowledge.106

这句话体现了学术界计算机科学家对“人为因素”的轻蔑态度，这种态度后来变得十分普遍。这关系到建立对计算机代码语义的自愿控制的可能性。从形式主义的角度来看，编程语言承诺了以代数式的方式随意为符号赋值的自由，从而建立了不受他人语言实践影响的独立性。

This comment exemplifies a dismissive attitude toward “human factors” that would become common among academic computer scientists. At stake was the potential to establish voluntary control over the semantics of computer code. Viewed through a formalist lens, programming languages promised the liberty to assign values to symbols at will in the manner of the algebraic let, thus establishing independence from the linguistic practices of other people.

这种对意义的个人主义态度与致力于使用逻辑数学方法分析系统相辅相成。在1962年的一篇文章中，约翰·麦卡锡提出了一种“计算的数学科学”，它将使“证明给定的程序能够解决给定的问题”成为可能。107其理念是提供一种比实践中常用的反复试验过程更为严格的软件开发方法。通过在特定输入上运行程序并检查输出来测试程序并不能令人满意，因为程序通常会在某些情况下运行，而在另一些情况下失败。程序证明将使代码的评估不再通过经验测试，而是通过对算法本身的数学分析，从而确保在所有情况下的可靠性。这一目标要求算法以形式化符号表示，因此与萨梅特倡导的自然语言方法相冲突。

This individualistic attitude toward meaning was consanguineous with a commitment to logicomathematical methods for analyzing systems. In a 1962 article, John McCarthy had proposed a “mathematical science of computation” that would make it possible “to prove that given procedures solve given problems.”107 The idea was to provide a more rigorous method of software development than the trial-and-error processes often used in practice. Testing programs by running them on particular inputs and checking the output was unsatisfactory since, quite commonly, procedures will work in some cases and fail in others. Program proofs would enable code to be evaluated not by empirical tests but through a mathematical analysis of the algorithm itself, thus ensuring reliability in all cases. This goal required algorithms to be expressed in a formal notation, and so it conflicted with the natural-language approach advocated by Sammet.

尽管事实证明，机械地解析英语语法比萨米特最初设想的要难，但形式主义的编程方法也带来了自身的难题。毕竟，人们必须能够理解代码，而为了达到这个目的，语言的诱惑力是难以抵挡的。1988年，当时在德克萨斯大学奥斯汀分校担任特聘教授的荷兰计算机科学家埃德斯格·W·迪杰斯特拉（Edsger W. Dijkstra）在一篇题为《论真正教授计算的残酷性》的文章中表达了他对这个问题的看法。科学。” 108他写道，理解新事物的一种常见方法是“我们试图通过隐喻和类比将新事物与旧事物、新奇事物与熟悉事物联系起来。” 109他认为，计算机呈现的情况如此新颖，以至于这种方法失效；“尽管我们可能用‘常识’来美化它，但我们过去的经验不再相关，类比变得过于肤浅，隐喻变得更具误导性而非启发性。” 110迪杰斯特拉延续了悠久的传统，将这些公认的思维方式与语言联系起来。“掌握一种彻底的新事物，”他继续说道，“相当于创造和学习一种无法翻译成母语的新外语。 ” 111

Although mechanically parsing English grammar proved harder than Sammet initially thought, formalist approaches to programming raised difficulties of their own. People, after all, must be able to understand code, and for this purpose it was hard to resist the siren call of language. In 1988, the Dutch computer scientist Edsger W. Dijkstra, then in an endowed chair at the University of Texas at Austin, expressed his views on the matter in the provocatively titled “On the Cruelty of Really Teaching Computing Science.”108 A common approach to understanding new things, he writes, is that “by means of metaphors and analogies we try to link the new to the old, the novel to the familiar.”109 Computers, he argues, present a situation so radically novel that this approach breaks down; “though we may glorify it with the name ‘common sense,’ our past experience is no longer relevant, the analogies become too shallow, and the metaphors become more misleading than illuminating.”110 Dijkstra continues a long tradition by linking these received ways of thinking to language. “Coming to grips with a radical novelty,” he continues, “amounts to creating and learning a new foreign language that can not be translated into one’s mother tongue.”111

实际上，创建这种新语言意味着从教学中消除隐喻、例子和其他现实世界的痕迹。例如，迪杰斯特拉讨论了算术，在这方面他听起来很像萨拉·波特 1835 年出版的《算术对话》中那个自以为是的大孩子。迪杰斯特拉写道：“不是教 2 + 3 = 5，而是把可怕的算术运算符‘加’精心伪装成‘和’，并且先给小孩子大量熟悉的例子，还有清晰可见的物体，如苹果和梨。” 112他称这是一种“愚蠢的传统”；他认为，用“可怕的”符号语言来教算术更好。113他建议在教授编程时也采取类似的简朴态度。 “为了训练新手程序员如何操作未解释的公式，”他建议，教授们应该将逻辑“更多地作为布尔代数来教授，让学生熟悉逻辑联结词的所有代数性质。为了进一步切断与直觉的联系，我们将布尔域的值{真，假}重命名为{黑，白}。” 114将逻辑术语转换为颜色的目的是消除任何暗示符号并非完全任意的暗示，从而防止学生受到诱导去思考意义。

In effect, creating this new language would mean eliminating metaphors, examples, and other traces of the real world from pedagogy. As an example, Dijkstra discusses arithmetic, in regard to which he sounds much like the presumptuous older child in Sarah Porter’s 1835 Conversations on Arithmetic. “Instead of teaching 2 + 3 = 5,” Dijkstra writes, “the hideous arithmetic operator ‘plus’ is carefully disguised by calling it ‘and’, and the little kids are given lots of familiar examples first, with clearly visible objects such as apples and pears.”112 He calls this a “silly tradition”; it is better, according to him, to teach arithmetic using the “hideous” language of symbols.113 He suggests a similar austerity in teaching programming. “In order to train the novice programmer in the manipulation of uninterpreted formulae,” he suggests, professors should teach logic “more as boolean algebra, familiarizing the student with all algebraic properties of the logical connectives. To further sever the links to intuition, we rename the values {true, false} of the boolean domain as {black, white}.”114 The point of switching from logical terms to colors is to remove any hint that the symbols are anything other than wholly arbitrary, and thus to protect students from being tempted into thinking in terms of meaning.

迪杰斯特拉对这种新的符号语言抱有远大的期望（这种期望一直以来都是如此）。他预测，计算技术终有一天会“超越其母学科——数学和逻辑，有效地实现莱布尼茨梦想的重要部分，即提供符号计算作为人类推理的替代方案”。115迪杰斯特拉认为，这种超越性面临着许多敌人，其中包括那些过于迷恋计算机可以让他们的生活更轻松的商界和军方，以及“所有软科学，而计算技术如今已成为这些软科学的某种跨学科避风港” 。116他还将自己的提议与人工智能 (AI) 领域区分开来，后者在 1988 年处于某种低谷：“试图用机器来模仿人类思维，我一直觉得这很愚蠢：我宁愿用它们来模仿更好的东西。” 117为了实现莱布尼茨的梦想（迪杰斯特拉特意大写了这个梦想），学生们必须学会将程序视为需要抽象推理的符号公式，而非机器所做之事的描述。这种严格的实践需要痛苦地驱除那些误导人们误以为代码与既有概念相关的意义、隐喻和类比。

Dijkstra’s hopes for this new symbolic language were (as such hopes always have been) lofty. Computing, he predicts, will one day “transcend its parent disciplines, mathematics and logic, by effectively realizing a significant part of Leibniz’s Dream of providing symbolic calculation as an alternative to human reasoning.”115 This transcendence, according to Dijkstra, has a number of enemies, among them business and military communities too enamored with the idea that computers can make their lives easier, as well as “all soft sciences for which computing now acts as some sort of interdisciplinary haven.”116 He also differentiates his proposal from the field of artificial intelligence (AI), which in 1988 was at something of a nadir: “The effort of using machines to mimic the human mind has always struck me as rather silly: I’d rather use them to mimic something better.”117 To realize Leibniz’s Dream (which Dijkstra imperiously capitalizes), students must learn to think of programs less as descriptions of things machines do than as symbolic formulae that must be reasoned about abstractly. This rigorous practice would require a painful exorcism of the meanings, metaphors, and analogies that fooled people into thinking code had anything to do with received notions.

到 1988 年，当 Dijkstra 写作时，这种对代码的严肃看法已经在很大程度上让位于当时所谓的“用户友好性”。118与Dijkstra 相比，Knuth 对语言的态度截然相反，他发展了文学编程的概念，其中代码包含对其自身的散文解释。119 Grace Hopper 继续倡导考虑编程语言部署的社会背景的重要性。120 Sandy Payette 注意到 Dijkstra 将 Hopper 的方法描述为“软”的，其中暗含着性别歧视；对人的关注被视为女性化，因此与科学无关。121虽然这些“软”问题在学术界继续被贬低，但在私营部门，它们很难被忽视。尤其是在 20 世纪 70 年代末苹果电脑公司成立之后，计算机行业为创造力和美学腾出了空间。创造力和美学被视为取悦客户、开拓新市场的一种方式，因为它鼓励人们将计算机融入到生活的方方面面。在商业环境中，Dijkstra 所倡导的“残酷”行不通——开发畅销的软件意味着构建为人们服务的系统，而为了实现这一目标，与现有语言彻底决裂是不可能的。

By 1988, when Dijkstra was writing, this austere view of code had already lost a great deal of ground to what was, by then, coming to be called “user friendliness.”118 Taking much the opposite approach to language compared to Dijkstra, Knuth developed the idea of literate programming, in which code contains a prose explanation of itself.119 Grace Hopper continued to advocate for the importance of considering the social contexts in which programming languages were deployed.120 Sandy Payette has noted an undercurrent of sexism in Dijkstra’s description of Hopper’s approach as “soft”; a concern with people was treated as feminine and thus extrinsic to science.121 While such “soft” matters continued to be devalued in the academy, they were harder to ignore in the private sector. Especially after the founding of Apple Computer in the late 1970s, the computer industry made room for creativity and aesthetics, which were valued as a way of pleasing customers and opening new markets by encouraging people to incorporate computers into more aspects of their lives. In commercial contexts, the “cruelty” Dijkstra advocated would not do—making software that sold meant constructing systems that worked for people, for which purpose a radical break with existing languages was not on the table.

Ada 是一种明确优先考虑“软”问题的编程语言。从 1977 年开始，美国国防部举办了一场竞赛，以开发一种用于其运营的标准编程语言。获奖作品由 Jean Ichbiah 领导的团队开发，他曾在法国公司 CII Honeywell Bull 工作。与许多现代编程语言一样，Ada 是 ALGOL 的后代，但它更加强调“对人类程序员的关注”。122正如一份基本原理文件所述，Ada 的设计基于这样一种信念：“如果我们的直觉能够依赖于传达程序逻辑结构的文本形式，我们对程序的理解就会大大简化”；因此，Ada 语言“主要考虑可读性和可教性”。123受Barbara Liskov 的 CLU 编程语言的影响，Ada 将代码划分为称为“包”的模块，其中包括用于存储数据的过程和结构。124这种方法（现在称为面向对象编程的前身）的目的不是为了数学证明，而是为了通过更容易地分配任务和传达复杂系统特定部分的用途来促进团队合作。

One programming language that explicitly privileged “soft” matters was Ada. Starting in 1977, the US Department of Defense ran a contest to develop a standard programming language for its operations. The winning entry was developed by a team led by Jean Ichbiah, who worked for the French company CII Honeywell Bull. Like many modern programming languages, Ada is a descendant of ALGOL, but it placed a heightened emphasis on “concern for the human programmer.”122 The design of Ada, as a rationale document states, rests on the belief “that our understanding of programs can be greatly simplified if our intuition is able to rely on textual forms that convey the logical structure of the program”; as a result, the Ada language gives “major consideration to readability and teachability.”123 Influenced by Barbara Liskov’s CLU programming language, Ada divided code into modules called “packages” that included both procedures and structures for storing data.124 This approach (a precursor to what is now called object-oriented programming) was tailored less toward mathematical proof than toward facilitating teamwork by making it easier to divvy up tasks and communicate the purposes of specific parts of a complex system.

正如“Ada”这个名字的选择所表明的那样，程序员们为他们的职业汇集历史先例典范。拜伦勋爵的女儿、查尔斯·巴贝奇计算机引擎注释的作者艾达·洛夫莱斯，一方面代表了女性在计算机时代早期所扮演的角色，另一方面也表明了需要用其他东西来补充数学方法。在洛夫莱斯被尊崇的过程中，一位重要人物是贝蒂·亚历山德拉·图尔，她于 1992 年出版了洛夫莱斯部分信件的详细注释版，图尔与 Ada 编程语言的一些开发人员合作编辑了该版本。图尔认为，洛夫莱斯将数学敏锐性与诗意想象力相结合，使她成为数字时代的合适偶像。图尔写道，现代数学既需要“客观性、观察和实验等数字技能，也需要“想象力、可视化和运用隐喻等模拟技能”。125正如 Toole 所报告的，对 Ada 程序员的调查发现，其最重要的属性是“连通性，即使用该语言工作的团队进行交流的能力和便利性。” 126 Ada 的设计延续了 Lovelace 等十九世纪数学家的重点，即在机械方法的优势与有机培养的直觉以及清晰优雅地解释想法的能力之间取得平衡。

As the choice of the name “Ada” indicates, programmers were by this point assembling a canon of historical precedents for their profession. Ada Lovelace, Lord Byron’s daughter and the writer of notes on Charles Babbage’s computing engines, came to stand in part for the role women played in the early days of the computer age and in part for the need to supplement mathematical methods with something else. An important figure in the canonization of Lovelace was Betty Alexandra Toole, who in 1992 published a heavily annotated edition of some of Lovelace’s letters that Toole edited in collaboration with some of the developers of the Ada programming language. According to Toole, it is Lovelace’s combination of mathematical acumen with poetic imagination that makes her an appropriate icon for the digital age. Modern mathematics, Toole writes, requires both “digital skills such as objectivity, observation and experimentation, and analogue skills such as imagination, visualization and the use of metaphor.”125 As Toole reports, a poll of Ada programmers found that its most important attribute was “connectivity, the ability and ease of teams of people working in the language to communicate.”126 The design of Ada continued the focus of nineteenth-century mathematicians like Lovelace on balancing the advantages of mechanical methods with an organically cultivated intuition and an ability to explain ideas clearly and elegantly.

尽管人们普遍认识到沟通的重要性，但“软”方法却因缺乏严谨性而受到抵制。1980 年，牛津大学计算机科学家 CAR Hoare 在获得图灵奖的演讲中对 Ada 提出了著名的批评，Hoare 以快速排序算法的发明者而闻名。通过颠倒皇帝的新衣的故事，Hoare 暗示 Ada 有可能变成一堆没有皇帝的衣服。虽然他直接指出该语言太复杂而不值得信赖，但这种比较也表明了它的肤浅：“小玩意和光鲜亮丽的东西凌驾于安全和经济的基本考虑之上。” 127他认为，Ada 包含一些“危险”的特性，这些特性会导致错误，因为它们会在程序的不同部分之间产生令人困惑的相互依赖关系。128 Dijkstra也对早期编程语言中常见的go to语句提出了类似的批评。 129与 Dijkstra 和 McCarthy 一样，Hoare 力求以可论证的确定性来确保代码的正确性——即“任何源语言程序在逻辑上都不可能导致计算机失控”。130一些类似的技术已成为主流软件工程的一部分，包括强类型和内存保护。但在实践中，代码的行为有时确实难以预测，其正确性并不总是能够用数学上的确定性来判断。

In spite of the general recognition of the importance of communication, “soft” approaches faced pushback for a perceived lack of rigor. A prominent criticism of Ada appeared in the 1980 Turing Award lecture of the Oxford computer scientist C. A. R. Hoare, best known as the inventor of the Quicksort algorithm. Inverting the story of the Emperor’s New Clothes, Hoare implies that Ada is in danger of becoming a pile of clothes without an emperor. While his immediate point is that the language is too complicated to rely on, the comparison also suggests superficiality: “Gadgets and glitter prevail over fundamental concerns of safety and economy.”127 Ada, he argues, contains “dangerous” features that lead to errors because they create confusing interdependencies between distant parts of programs.128 Dijkstra had made a similar criticism of the go to statements that were common in early programming languages.129 Like Dijkstra and McCarthy, Hoare sought to ensure the correctness of code with demonstrative certainty—to make it “logically impossible for any source language program to cause the computer to run wild.”130 Some techniques along these lines have become a part of mainstream software engineering, including strong typing and memory protection. But in practice, code did sometimes behave unpredictably, and correctness could not always be judged with mathematical certainty.

证明程序正确性的想法最终在 Haskell 编程语言中得以体现，该语言是在 1987 年国际会议上构思出来的基于函数式编程，于 1990 年首次发布。131 Haskell 以几年前去世的宾夕法尼亚州立大学数学教授 Haskell B. Curry 的名字命名。Curry 和 William Alvin Howard 一起发现了一种在计算机程序和数学证明之间转换的方法，后来被称为 Curry-Howard 对应关系。132这种对应关系提供了一种证明程序将具有某些行为的方法。然而，这种方法只有在语言是纯函数式的情况下才有效，这意味着代码单元不能影响其正式定义的输出之外的任何东西。这种限制要求 Haskell 将与其他系统（如用户界面）的交互封装在一个抽象层中——称为monad ，这显然不是对莱布尼茨形而上学的引用——以防止外界损害程序的逻辑纯度。

The idea of proving programs correct culminated in the Haskell programming language, which was conceived at the 1987 International Conference on Functional Programming and first released in 1990.131 Haskell was named in honor of Haskell B. Curry, a professor of mathematics at Pennsylvania State University who had died a few years before. Along with William Alvin Howard, Curry had discovered a method of converting between computer programs and mathematical proofs that came to be known as the Curry–Howard correspondence.132 This correspondence provides a way to prove that a program will have certain behaviors. This approach only works, however, if the language is purely functional, meaning that a unit of code cannot affect anything beyond its formally defined output. This limitation requires Haskell to encapsulate interactions with other systems (such as user interfaces) within a layer of abstraction—called, in what is apparently not a reference to Leibniz’s metaphysics, a monad—to prevent the outside world from compromising the logical purity of programs.

Haskell 仍在积极开发中，并拥有一个热情的用户社区，但它的普及程度远不及 C++、Python 和 Go 等语言，后三者都包含无法通过严格证明的元素。133即使在 Haskell 中，形式规则也不能（正如 Church 和 Turing 所证明的那样）完全决定正确的处理方式。典型的编程体验不像 Dijkstra 对未解释公式的分析，而更像 Toole 在“数字”和“模拟”之间的平衡。软件开发的很大一部分涉及沟通：编写其他团队成员可以阅读的代码，遵循内部风格，以合理的方式分配工作，处理从过去员工那里继承的“遗留”系统，以及为未来的维护者和用户记录新系统。

Haskell is still actively developed and has an ardent community of users, but it is far less widespread than such languages as C++, Python, and Go, which all contain elements that are not amenable to strict proofs.133 Even in Haskell, formal rules cannot (as Church and Turing proved) wholly determine the right way of proceeding. The typical experience of programming is less like Dijkstra’s analysis of uninterpreted formulae than like Toole’s balance between “digital” and “analogue.” A large part of software development involves communication: writing code that other team members can read, following house styles, portioning out work in sensible ways, dealing with “legacy” systems inherited from past employees, and documenting new systems for future maintainers and users.

这些问题被纳入了计算机科学学科。麻省理工学院的计算机科学家 Harold Abelson、Gerald Jay Sussman 和 Julie Sussman 在他们颇具影响力的 1985 年教科书《计算机程序的结构和解释》中认为“程序必须写出来给人看，只是偶尔给机器执行”；他们不再强调“算法的数学分析和计算基础”，而更倾向于“用于控制大型软件系统智力复杂性的技术”，包括对风格和美学的欣赏。134 值得注意的是，这本书引用了约翰·洛克《人类理解论》中的一段题词，表明其关注抽象的认知行为。135现代编程语言远非 ALGOL 会议上设想的纯形式“算法语言”，而是复杂的野兽，它们服务于多种用途，并借鉴了各种影响，有些是逻辑数学的，有些则不是。

Such matters were incorporated into the academic discipline of computer science. In their influential 1985 textbook Structure and Interpretation of Computer Programs, the MIT-affiliated computer scientists Harold Abelson, Gerald Jay Sussman, and Julie Sussman take the position that “programs must be written for people to read, and only incidentally for machines to execute”; they de-emphasize “the mathematical analysis of algorithms and the foundations of computing” in favor of “the techniques used to control the intellectual complexity of large software systems,” including an appreciation for style and aesthetics.134 It is telling that the book includes an epigraph from John Locke’s Essay Concerning Human Understanding, indicating a focus on the cognitive act of abstraction.135 Far from the purely formal “algorithmic language” envisioned at the ALGOL conferences, modern programming languages are complex beasts that serve a range of purposes and draw on a range of influences, some logicomathematical and some not.

因此，算法成为该学科的众多主题之一。20 世纪 60 年代初，计算机科学家将算法视为程序的同义词； CACM中“算法”部分的框架强化了这一点。136到了 20 世纪 80 年代，算法和程序已经分道扬镳。艾伦·J·佩利斯 (Alan J. Perlis) 在其《计算机程序的结构和解释》的前言中，将算法视为程序应该追求的理想：

Algorithms thus became one topic among many within the discipline. In the early 1960s, computer scientists treated algorithm as a synonym for program; the framing of the “Algorithms” section in CACM reinforced this.136 By the 1980s, algorithms and programs had diverged. In his forward to Structure and Interpretation of Computer Programs, Alan J. Perlis holds up the algorithm as an ideal to which programs should aspire:

在我们编写的程序中，有些程序（但数量远远不够）执行精确的数学函数，例如排序、求数列的最大值、确定素数或求平方根。我们称这类程序为算法，人们对它们的最优行为了解甚多，尤其是在执行时间和数据存储需求这两个重要参数方面。137

Among the programs we write, some (but never enough) perform a precise mathematical function such as sorting or finding the maximum of a sequence of numbers, determining primality, or finding the square root. We call such programs algorithms, and a great deal is known of their optimal behavior, particularly with respect to the two important parameters of execution time and data storage requirements.137

即使只有少数精英程序能够达到真正算法的地位，我们仍然可以将算法理论应用于程序的某些部分——执行明确定义任务的程序，这些程序与使用它们的更广泛的软件系统隔离开来。将算法从程序中抽象出来，使计算机科学家能够以至少部分 ALGOL 创建者所设想的严谨性来分析它们，即使这种严谨性无法扩展到实际的生产代码。这种严谨性过去是、现在仍然是真实的——它极大地提高了软件的可靠性和效率——但它是有代价的。算法理论，正如它在计算机科学中被教授的那样，在计算和人类生活世界之间设置了一道障碍，其全部影响在几十年后才会显现出来。

Even if only an elite few programs could achieve the status of true algorithms, one could still apply the theory of algorithms to parts of programs—to procedures that perform clearly defined tasks, considered in isolation from the broader software systems in which they are used. Abstracting algorithms from programs enabled computer scientists to analyze them with at least some of the rigor ALGOL’s creators envisioned, even when that rigor could not extend to actual production code. This rigor was and is genuine—it contributed greatly to the reliability and efficiency of software—but it came at a cost. The theory of algorithms, as it came to be taught in computer science, threw up a barrier between computation and the world of human life whose full implications would not become apparent for decades.

问题与解决方案

Problems and Solutions

在对技术的批判性研究中，人们一直在讨论定位算法的难度。138它们是在代码中、可执行文件中，还是在运行的电路中？如何区分系统中哪些部分是算法，哪些不是？一个答案是，算法根本不在计算机中——它们存在于书籍和学术期刊的页面上。计算机科学学位课程通常要求学习“数据结构和算法”课程，学生将在其中学习一系列众所周知的计算过程及其推理技巧。ALGOL 项目旨在为这些过程提供标准符号，但这一目标并未实现：最早的计算机科学教科书之一，Alexandra I. Forsythe 和同事合著的《计算机科学：入门课程》（1969 年），使用流程图，而其他教科书则使用伪代码（一种半形式化的符号）（因作者而异）或真正的编程语言。139程序员需要熟记一些算法。例如，一个常见的面试问题就是详述一个用于确定网络是否包含循环的算法。学习算法很重要，不仅因为它们自身的实用性，还因为算法理论对于开发能够高效大规模运行的软件至关重要。

In critical studies of technology, there has been some discussion about the difficulty of locating algorithms.138 Are they in the code, in the executable files, or in the running circuitry? How can one tell which parts of a system are algorithms and which are not? One answer would be that algorithms are not in computers at all—they are on the pages of books and academic journals. Computer science degree programs typically require a course on Data Structures and Algorithms, in which students learn a canon of well-known computational procedures along with techniques for reasoning about them. The ALGOL project’s goal of providing a standard notation for these procedures did not obtain: one of the earliest CS textbooks, Alexandra I. Forsythe and colleagues’ Computer Science: A First Course (1969), uses flowcharts, while others use either pseudocode (a semiformal notation that varies from author to author) or real programming languages.139 Programmers are expected to know some algorithms by heart. A common job interview question, for instance, involves recounting an algorithm for determining whether a network contains a loop. Studying algorithms is important not just because of their individual utility but also because the theory of algorithms is crucial to developing software that can operate efficiently at scale.

希望学习机器学习和预测模型的学生有时会对他们在算法课程中遇到的内容感到惊讶。这些课程主要涵盖我们所谓的经典算法——人类设计的用于解决精确定义问题的程序。典型的问题是排序，它在教科书中随处可见，因为它阐明了算法理论中出现的许多问题。假设需要对一架书进行排序。一种简单的方法是交换所有顺序错误的相邻书籍对，然后重复此过程，直到所有书籍都排好序。Aho、Hopcroft 和 Ullman 在 1987 年版的教科书《数据结构和算法》中以基于 PASCAL 编程语言的符号介绍了这种方法，称为冒泡排序： 140

Students hoping to learn about machine learning and predictive modeling are sometimes surprised at what they encounter in algorithms courses. These courses largely cover what we might call classical algorithms—human-designed procedures for solving precisely defined problems. The paradigmatic problem is sorting, which is ubiquitous in textbooks because it illustrates a number of issues that arise in the theory of algorithms. Suppose that one has to sort a shelf of books. A simple approach would be to swap all pairs of adjacent books that are in the wrong order, then repeat until the books are all sorted. The 1987 edition of the textbook Data Structures and Algorithms by Aho, Hopcroft, and Ullman presents this method, which is called bubble sort, in a notation based on the PASCAL programming language:140

（1） 对于 i := 1至 n –1执行

（2）     对于 j := n 直到 i +1执行

（3）         如果 A [ j ].key < A [ j - 1] .key 则

（4）             交换（A [ j ]，A [ j -1]）

(1) for i := 1 to n–1 do

(2)     for j := n downto i+1 do

(3)         if A[j].key < A[j–1].key then

(4)             swap(A[j], A[j–1])

该过程使用两个for循环，它们重复执行其下方缩进的代码块。循环 (2) 以相反的顺序遍历数组的未排序部分，并在行 (3) 和 (4) 中交换相邻的乱序项。循环 (1) 重复该过程所需的次数。请注意，只要指定所需的顺序（由key表示，它为每个项分配一个数值），该算法就可以用于对任何类型的数据进行排序。然而，当应用于大量输入时，此方法速度极慢，因此它被当作糟糕算法的例子来教授。对更高效排序的渴望导致了许多不太明显的排序程序的开发。最著名的是快速排序，它涉及根据项相对于任意选择的项的排名将项移动到一侧或另一侧，然后对每一侧递归地执行相同的过程；这种经过大量研究的程序成为了优秀（即高效）算法设计的典范。

The procedure uses two for loops, which repeat the blocks of code indented beneath them. Loop (2) goes through the unsorted portion of the array in reverse order and, in lines (3) and (4), swaps adjacent items that are misordered. Loop (1) repeats that process the necessary number of times. Note that this algorithm can be used to sort any type of data whatsoever, provided that the desired order (represented by key, which assigns each item a numerical value) is specified. This method, however, is extremely slow when applied to large inputs, on account of which it is taught as an example of a bad algorithm. The desire for more efficient sorting led to the development of a number of less obvious sorting procedures. The most famous is Quicksort, which involves moving the items to one side or the other based on their ranking relative to an arbitrarily chosen item, then performing the same procedure recursively for each side; this much-studied procedure became a paradigm of good (meaning efficient) algorithm design.

虽然计算机科学家从未对算法达成共识，但算法教科书中的程序确实具有一些共同特征。其一，用 Knuth 的话来说，这些程序是有限的、确定的和有效的，也就是说，它们由最终终止的明确定义的操作序列组成，每个操作都可以在有限的时间内完成。141正如 Aho、Hopcroft 和 Ullman 指出的那样，一个程序是否满足这些标准是有争议的，因为“一个人清楚的事情另一个人可能不清楚，而且通常很难严格证明一条指令可以在有限的时间内执行。” 142虽然很难证明一个程序是一种算法，但更容易证明有些程序不是。例如，假设我们想验证 Michael Stifel 在计算机时代前四个多世纪的成果，即。我们可能会想出这样的东西（伪代码）：

While computer scientists never reached a consensus definition of algorithm, the procedures in algorithms textbooks do share some common features. One is that the procedures are, in Knuth’s terms, finite, definite, and effective—that is, they consist of an eventually terminating sequence of clearly defined operations that can each be completed in a finite amount of time.141 As Aho, Hopcroft, and Ullman point out, whether a procedure meets these criteria can be debatable, since “what is clear to one person may not be clear to another, and it is often difficult to prove rigorously that an instruction can be carried out in a finite amount of time.”142 Although it may be hard to prove that a procedure is an algorithm, it is easier to show that some procedures are not. For instance, suppose we wanted to verify Michael Stifel’s result, from over four centuries before the computer era, that . We might come up with something like this (in pseudocode):

但这不是一个有效的程序，至少如果我们把它解释为我们提出的问题的解决方案的话。问题在于，正如施蒂费尔惊愕地观察到的那样，这些根的值只能近似。143我们可以根据需要计算任意多位近似值（= 7.0710678……），但比较a和b的所有数字需要无限多的步骤。这一点很微妙，但它说明了算法与其他形式的数学推理（例如施蒂费尔用来得出该方程的几何方法）之间的重要区别。经典算法存在于图灵机的世界中，其中运算的基本对象是数字，而不是数字。144因此，它们被无限的幽灵所困扰，被程序永远持续下去的可能性所困扰，指定的停止标准永远不会实现。145

But this is not an effective procedure, at least if one interprets it as a solution to the problem we have set out. The issue is that, as Stifel observed with such consternation, the values of those roots can only be approximated.143 We can compute as many digits of those approximations as we like ( = 7.0710678 . . .) but it would take infinitely many steps to compare all the digits of a and b. This point is subtle, but it illustrates an important difference between algorithms and other forms of mathematical reasoning, such as the geometric method Stifel used to arrive at that equation. Classical algorithms inhabit the world of Turing Machines, in which the fundamental objects of operation are digits, not numbers.144 They are thus haunted by the specter of infinity, by the possibility that a procedure will go on forever, the specified stopping criterion never coming true.145

与此相关的一点是，算法只能以某些有限的方式与其他系统交互——即通过输入和输出。正如 Knuth 所说，算法必须有零个或多个“取自指定对象集合”的输入，以及一个或多个“与输入具有指定关系”的输出。146指定这种关系并不意味着提前知道输出——在很多情况下，这会使计算机冗余——但我们仍然必须以某种精确的方式定义输出，就像方程式无论我们是否知道它的根一样。在某些情况下，算法包含随机因素或仅提供近似解，但所需的输出仍然必须指定。从这个意义上讲，算法仅涵盖计算机所做工作的一小部分；特别是，需要从环境持续反馈的系统，例如机器人控制系统和用户界面循环，从技术上讲并非算法。虽然计算机科学家早已认识到这一概念的局限性，并试图扩展算法的概念以涵盖更具动态性的流程，但为了使算法理论发挥作用，需要某种程度的狭窄定义。147复杂性理论尤其只有在输入明确限定的情况下才有效；否则，结果可能取决于从气温到坐在键盘前的人的突发奇想等许多难以处理的因素。对于输出，模糊定义更容易避免，但如果我们希望证明算法的正确性，精确指定的目标是至关重要的。

A related point is that algorithms are only able to interact with other systems in certain limited ways—namely, through the input and output. As Knuth puts it, algorithms must have zero or more inputs “taken from specified sets of objects” and one or more outputs that “have a specified relation to the inputs.”146 Specifying this relation does not mean knowing the output ahead of time—in many cases, that would make the computer redundant—but we must still define the output in some precise way, as an equation defines its roots whether we know them or not. In some cases, the algorithm includes a random factor or supplies only an approximate solution, but the desired output must still be specified. Algorithms in this sense cover only a small slice of the things computers do; in particular, systems that require continuous feedback from an environment, such as robot control systems and user interface loops, are not technically algorithmic. While computer scientists have long recognized the limits of this conception and sought to expand the concept of algorithm to include processes that are more dynamic, some narrowness of definition is needed to make the theory of algorithms tick.147 Complexity theory, in particular, only works when that input is clearly circumscribed; otherwise, the outcome could depend on any number of intractable factors ranging from the air temperature to the whims of the person sitting at the keyboard. It is easier to get away with vagueness as to the output, but a precisely specified goal is crucial if we hope to prove an algorithm correct.

如何确定这个目标是一个复杂的问题。阿霍、霍普克罗夫特和厄尔曼宣称：“成功的一半在于知道要解决什么问题。” 148他们的教科书建议采用两阶段方法：首先准确定义问题，然后找出解决问题的算法。其他人则倾向于更灵活的方法。在 1968 年的北约会议上，迪杰斯特拉认为，不完整的问题规范比完整的规范更好，因为它们更容易开发可在其他情况下重复使用的通用解决方案。149这句话指出了算法的另一个特点，克努斯没有在他的定义中包括这个特点，而马尔可夫包括了：通用性。通常，算法不是要解决问题的单个实例，而是要能够在尽可能广泛的输入范围内重复使用。另一本早期教科书曾这样写道：“虽然人们可以很容易地学会如何解决各种类型的问题，但要识别看似截然不同的问题的共同结构并开发出解决这些问题的通用方法却更加困难。对如何解决一类问题的精确描述被称为算法。” 150就其声称的通用性而言，古典算法继承了韦达在 16 世纪 90 年代制定的帝国计划：就像他的方程求解定理一样，它们应该征服数百个问题，而不是逐个攻克它们。

Just how to determine this goal is a complex matter. “Half the battle,” Aho, Hopcroft, and Ullman declare, “is knowing what problem to solve.”148 Their textbook suggests a two-phase method: first define the problem precisely, then figure out an algorithm to solve it. Others have preferred a more flexible approach. At the 1968 NATO conference, Dijkstra argued that incomplete problem specifications were better than complete ones because they made it easier to develop general solutions that could be reused in other cases.149 This statement points to another characteristic of algorithms that Knuth does not include in his definition but that Markov does: generality. Typically, algorithms are not meant to solve only a single instance of a problem but to be reused with as wide a range of inputs as possible. Another early textbook puts it this way: “Although one may easily learn how to solve problems of many types, it is more difficult to recognize the common structure of problems which are seemingly quite different and to develop general methods for their solution. A precise description of how to solve a class of problems is called an algorithm.”150 To the extent that they claim generality, classical algorithms carry on the imperial program Viète enacted in the 1590s: like his equation-solving theorems, they are supposed to conquer problems by the hundreds rather than attacking them one by one.

诸如快速排序之类的经典算法不应与后来与“算法”一词相关的预测模型相混淆。算法教材中编写的程序大多与统计学无关，也不一定旨在预测任何事情。经典的算法也不同于它所计算的函数，即对给定输入的输出应如何定义的规范。这种区别有时在批判性学术研究中被忽视，但它对于理解经典算法的学科实践至关重要。151通常，有多种算法通过不同的方式计算相同的函数，因此就输出而言，它们可以互换。例如，只要排序没有平局，标准排序算法（例如冒泡排序、归并排序和快速排序）在给定相同输入的情况下都会产生完全相同的输出；它们的主要区别在于速度和内存使用情况。152相反，标准排序算法不会假设元素的顺序，而是生成用户指定的任何顺序。153经典算法主要处理如何计算事物，而不是计算什么，并且至少在理想情况下，人们可以应用它们而不必担心它们的细节，就像无数程序员使用 Python 的list.sort()方法时无需询问它采用的是哪种排序算法一样。 （事实上​​，它使用了 Tim Peters 开发的一种名为Timsort的算法。）

Classical algorithms such as Quicksort should not be conflated with the predictive models that later came to be associated with the word algorithm. The procedures compiled in algorithms texts are mostly unconnected to statistics, nor are they necessarily intended to predict anything. A classical algorithm is also distinct from the function it computes, meaning the specification of what the output should be for a given input. This distinction is sometimes overlooked in critical scholarship, but it is crucial for understanding the disciplinary practices that attend classical algorithms.151 Often there are multiple algorithms that compute the same function by different means and that are thus, as far as the output goes, interchangeable. For instance, as long as there are no ties in the ranking, standard sorting algorithms such as bubble sort, merge sort, and Quicksort will all produce exactly the same output, given the same input; they differ primarily in their speed and memory use.152 Conversely, standard sorting algorithms make no assumptions about what order items should be in, instead producing whatever ordering the user specifies.153 Classical algorithms deal primarily with how to compute things, not with what to compute, and, at least ideally, one can apply them without having to worry about their details, just as countless programmers have used Python’s list.sort() method without needing to ask which sorting algorithm it employs. (It uses, as it so happens, an algorithm developed by Tim Peters called Timsort.)

但是，这种可互换性只有在问题被精确定义的情况下才有效。排序问题通常足够精确，但其他问题可能要复杂得多。后来被称为“Dijkstra 算法”的程序就是一个很好的例子。Dijkstra 在 1959 年的一篇文章中首次发表了该程序，它可以找到两点之间的最佳路径。他最初的应用是寻找荷兰城市之间的路线；很久以后，无论公平与否，这种算法都被誉为 GPS 导航的基础。154 Dijkstra算法借鉴了图论的方法，图论是 20 世纪初发展起来的一种网络数学理论（图 5.3）。从这个意义上讲，图由一组节点或顶点以及连接节点对的边组成。在 Dijkstra 的公式中，边也有定义的长度。 Dijkstra 的方法利用了这样一个事实：如果图中的一条给定路径是最短的，那么该路径的任何子路径也必然是从其起点到终点的最短路径。（Dijkstra在文章中没有将他的程序称为算法；他称之为“过程”和“解决方案” 。155）大多数其他常见的寻路算法都是 Dijkstra 程序的变体；例如，有时用于引导视频游戏角色运动的 A* 算法增加了一种启发式算法，使搜索偏向于朝着目标大致方向的移动。156

This interchangeability only works, however, to the extent that the problem is precisely defined. Sorting problems are typically precise enough, but other problems can be much messier. A good illustration is the procedure that came to be known as “Dijkstra’s algorithm.” This procedure, which Dijkstra first published in a 1959 article, finds an optimal path between two points. His original application was to find routes between cities in the Netherlands; much later, this algorithm came to be celebrated, fairly or not, as the foundation of GPS navigation.154 Dijkstra’s algorithm draws on the methods of graph theory, a mathematical theory of networks that was developed in the early twentieth century (figure 5.3). In this sense, a graph consists of a set of nodes, or vertices, along with edges that connect pairs of nodes. In Dijkstra’s formulation, the edges also have defined lengths. Dijkstra’s method makes use of the fact that if a given path through a graph is the shortest one possible, any subpath of that path must also be the shortest possible from its starting point to its end. (Dijkstra does not refer to his procedure as an algorithm in the article; he calls it a “process” and a “solution.”155) Most other common pathfinding algorithms are variants of Dijkstra’s procedure; for instance, the A* algorithm, which is sometimes used to guide the motion of video game characters, adds a heuristic that biases the search toward moves in the general direction of the goal.156

从数学角度来看，很难否认 Dijkstra 算法的正确性：它确实找到了一条最优路径，这一点可以通过数学归纳法证明。但应用这样的算法需要表示这个问题被看作是一个网络，而其中涉及的选择可能会对现实世界产生影响。2017年，新泽西州莱奥尼亚的居民注意到他们居住的城镇交通流量突然增加，甚至到了完全拥堵的地步。157事实证明，施工导致乔治华盛顿大桥周围交通拥堵，谷歌地图和 Waze 等导航应用程序让汽车绕道而行。虽然局外人无法确切知道这些程序究竟使用了哪些寻路算法，但算法本身对这个问题的重要性远不如问题的定义方式。为了采用基于网络的寻路算法算法首先必须选择构成节点的位置。对于 Dijkstra 来说，这些位置是城市，而在街道地图上，它们通常是交叉路口。其次，必须决定为边分配哪些值。Dijkstra 将这些值称为“长度”，这表明它们代表物理距离，但还有许多其他方法可以标记边的相对可取性。导航应用程序通常会考虑一些因素，例如限速和当前交通状况，但在新泽西州的案例中，它们没有考虑为当地社区规划特定区域的路线成本。

From a mathematical perspective, it is hard to deny that Dijkstra’s algorithm is correct: it does indeed find an optimal path, as one can prove by mathematical induction. But applying such an algorithm requires representing the problem as a network, and the choices involved can make real-world differences. In 2017, residents of Leonia, New Jersey, noticed a sudden increase in traffic through their residential town, to the point of total gridlock.157 As it turned out, construction had led to backups around the George Washington Bridge, and navigation apps such as Google Maps and Waze were sending cars through side streets as an alternate route. While outsiders cannot know exactly which pathfinding algorithms these programs were using, the algorithms themselves are less important to this issue than how the problem was defined. To employ a network-based pathfinding algorithm, one must, first of all, choose which locations constitute the nodes. For Dijkstra, these were cities, whereas on a street map they are typically intersections. Second, one must decide what values to assign to the edges. Dijkstra called these values “lengths,” which suggests that they represent physical distances, but there are numerous other ways one could mark the relative desirability of edges. Navigation apps typically account for some factors such as speed limits and current traffic, but in the New Jersey case, they did not consider the cost of routing cars through a particular area for the local community.

这一事件说明了古典范式所带来的危险和机遇。158通过将应用程序的复杂细节包含在形式定义的输入和输出中，该范式为在稀薄的数学领域中进行严格研究计算逻辑打开了大门，尽管其更广泛的含义仍然（也必然如此）深不可测且尚未完全理解。IBM 2018 年的一段宣传视频以惊人的直率阐述了形式化的逻辑：“研究人员正在通过用工程术语定义人类价值观来帮助人工智能系统理解人类价值观。” 159人类价值观是否真的可以用工程术语来定义都是值得怀疑的。但算法思维要求构建与词语的传统含义并存的形式定义。从技术角度来看，这些定义是否真正捕捉到了它们想要体现的价值观，其次才是算法的正确性和效率，而这才是最终决定算法在实施时效果如何的因素。

This incident illustrates the dangers and the opportunities created by the classical paradigm.158 By containing the messy details of the application within the formally defined input and output, the paradigm opens computational logic to rigorous study within a rarefied mathematical realm even as its broader implications remain (as they must be) unfathomably complex and incompletely understood. An IBM promotional video from 2018 states the logic of formalization with striking bluntness: “Researchers are helping AI systems understand human values by defining them in engineering terms.”159 Whether human values can even in principle be defined in engineering terms is questionable. But algorithmic thinking demands the construction of formal definitions that exist alongside the conventional meanings of words. From a technical point of view, whether these definitions really do capture the values they are meant to embody is secondary to the correctness and efficiency of the algorithms, which is what ultimately determines how well they will work when implemented.

这种脱节在多大程度上成为一个问题，取决于诸如理念和价值观之类的东西究竟如何转化为这些正式定义。160虽然这些问题超出了算法理论的正式范围，但它们不容忽视。正如沃伦·萨克所指出的，计算系统需要修辞：它们的设计者必须说服人们接受他们选择的解决问题的特定方式。161在学术计算机科学中，这种修辞大量借鉴了二元论的知识观，这种观点随着 19 世纪逻辑数学形式主义的兴起而开始出现在符号方法中。一个典型的例子是备受讨论的算法，它是原始谷歌搜索引擎的关键组成部分：PageRank。162

The extent to which this disconnect is a problem hinges on how, exactly, things like ideas and values come to be translated into those formal definitions.160 While such questions fall outside the official scope of the theory of algorithms, they cannot be ignored. As Warren Sack has pointed out, computational systems require rhetoric: their designers must persuade people to accept the particular way they have chosen to frame the problem.161 In academic computer science, this rhetoric borrows heavily from the dualistic view of knowledge that began to attend symbolic methods with the rise of logicomathematical formalism in the nineteenth century. An illustrative case is the much-discussed algorithm that formed a key part of the original Google search engine: PageRank.162

PageRank 的目的是为网页分配数值分数，用于对搜索结果进行优先排序。（需要明确的是，PageRank 是一种排名算法，而非排序算法；实际上，必须使用像 Quicksort 这样的独立程序来对项目进行排序。）1998 年，谷歌创始人拉里·佩奇 (Larry Page) 还是斯坦福大学的研究生时，他和他的同事在一份名为《PageRank 算法》的白皮书中解释了他们的方法。引文排名：让网络井然有序”的论文，以及一篇更广泛地描述谷歌搜索引擎原型的期刊文章和两份专利申请中都提到了这一点。163白皮书没有花太多时间解释算法本身，而是详细阐述了其用于网页排名的原理。在此过程中，该论文以一种现代算法思维的典型方式，努力解决几个世纪以来数学家们用符号替代文字时反复出现的概念问题。

The purpose of PageRank is to assign web pages numerical scores that may be used to prioritize search results. (To be clear, PageRank is a ranking algorithm, not a sorting algorithm; a separate procedure such as Quicksort must be used actually to arrange the items in order.) In 1998, when the Google founder Larry Page was a graduate student at Stanford, he and his collaborators explained their method in a white paper titled “The PageRank Citation Ranking: Bringing Order to the Web” as well as in a journal article that more broadly describes the prototype of the Google search engine and in two patent applications.163 The white paper spends less time on explaining the algorithm itself than on expounding a rationale for its use as a ranking of web pages. In doing so, the paper grapples, in a way characteristic of modern algorithmic thinking, with the conceptual problems that have arisen time and again over the centuries when mathematicians have traded words for symbols.

PageRank 算法建立在传统的引文分析基础之上，通过计算反向链接（即有多少其他页面链接到给定页面）对网页进行评分。164与简单的引文计数方法相比，PageRank 算法更注重网络的结构。与 Dijkstra 算法一样，PageRank 算法根据图论的形式化问题，将网页表示为节点，将超链接表示为边。早在 20 世纪 50 年代，人们就常用基于节点连接的“中心性”度量来衡量图中节点的相对重要性；这些方法最初用于交流网络的社会计量研究。165 PageRank算法将这种网络分析与引文计数相结合。它不是平等对待所有连接，而是赋予来自 PageRank 值较高的页面的反向链接更高的权重。几十年前，Leo Katz 和 Phillip Bonacich 以及大约同一时期的 Jon M. Kleinberg 都提出了类似的想法。166使用 PageRank 对链接进行加权会导致定义循环：正如莱布尼茨的单子代表宇宙一样，每个页面的排名取决于网络上所有其他页面的排名。这个循环定义简化为一个方程组，可以使用线性代数的技巧求解，从而得出排名。

Building on the tradition of citation analysis, PageRank scores web pages by counting backlinks—that is, counting how many other pages link to a given page.164 Compared with simple citation counting methods, PageRank gives more significance to the network structure of the web. Like Dijkstra’s algorithm, it formalizes the problem in terms of graph theory, representing web pages as nodes and hyperlinks as edges. Going back to the 1950s, it had been common to measure the relative importance of nodes in a graph using various “centrality” measures based on their connections; these methods were initially used in sociometric studies of communication networks.165 PageRank combines this sort of network analysis with citation counting. Rather than treating all connections equally, it gives more weight to backlinks that come from pages that themselves have higher PageRanks. Similar ideas were proposed decades before by both Leo Katz and Phillip Bonacich and around the same time by Jon M. Kleinberg.166 Weighting the links by PageRank makes the definition circular: just as Leibniz’s monads each represent the universe, the rank of each page depends on the ranks of all the other pages on the web. This circular definition reduces to a system of equations that may be solved using techniques from linear algebra, thus producing the ranking.

Page 和他的白皮书合著者在摘要的开头阐述了他们开发这一指标的目的：“网页的重要性本质上是一个主观问题，取决于读者的兴趣、知识和态度。但关于网页的相对重要性，我们仍然可以客观地评价很多内容。” 167 PageRank 通过“客观且机械地”对网页重要性进行排名，使搜索引擎能够对显示给用户的结果进行优先排序。168这句话引出了 Lorraine Daston 和 Peter Galison 所说的“机械客观性”——机械化是一种将所有人类判断从实践中剔除的方式。169但仅仅使用机器并不能成为客观性的充分条件，原因很简单：计算机可以被编程为说出任何内容。（算法客观性的真正对手不是主观性，而是任意性。）因此，作者的任务是让读者相信他们的计算与现有的“重要性”概念具有足够强的相关性，值得作为一种衡量标准。

Page and his white paper coauthors state their purpose in developing this metric at the beginning of their abstract: “The importance of a Web page is an inherently subjective matter, which depends on the readers [sic] interests, knowledge and attitudes. But there is still much that can be said objectively about the relative importance of Web pages.”167 By ranking the importance of pages “objectively and mechanically,” PageRank enables the search engine to prioritize results that are shown to the user.168 This phrase invokes what Lorraine Daston and Peter Galison have called “mechanical objectivity”—mechanization as a way of taking all human judgment out of a practice.169 But merely using a machine cannot be a sufficient condition for objectivity for a simple reason: computers can be programmed to say anything. (The true antagonist of algorithmic objectivity is not subjectivity but arbitrariness.) The authors are thus tasked with convincing the reader that their calculation bears a strong enough relevance to the existing idea of “importance” to make it worth adopting as a measurement.

这一论证采取了从文字到符号的逐步推进的形式。首先，作者回顾了以前关于引文分析的研究，并说明了如何将该方法应用于网络链接以及学术引文。其次，他们认为简单的引文计数并不总是符合“我们常识中的重要性概念”，因为来自主要网站的链接比来自不知名网站的链接更重要。170在下一部分中，只有两句话，他们给出了“PageRank 的直观描述”：“如果一个页面的反向链接排名总和很高，那么该页面的排名就很高。” 171在这个口头定义之后，他们提出了一个符号方程来“形式化直觉”。172令B _u为链接到页面u 的页面集合，N _u为页面u上的链接数，c为常数。页面u的排名R可以定义如下：

This argument takes the form of a gradual progression from words to symbols. First, the authors review previous work on citation analysis and make a case for how this method could be applied to web links as well as to academic citations. Second, they argue that simple citation counting does not always correspond to “our common sense notion of importance,” since links from major websites are more important than links from obscure ones.170 In the next section, which consists of only two sentences, they state an “intuitive description of PageRank”: “a page has high rank if the sum of the ranks of its backlinks is high.”171 After this verbal definition, they present a symbolic equation that “formalizes the intuition.”172 Let B_u be the set of pages that link to the page u, N_u be the number of links on page u, and c be a constant. The ranking R of a page u may be defined as follows:

由于其基于直观的原理，该方程被认为构成了重要性的可计算定义。

Because of its grounding in an intuitive rationale, this equation is supposed to constitute a computable definition of importance.

接下来是一些回溯。虽然作者认为这符合“常识”，但这个初始方程有一个缺陷：页面组之间可能互相引用，从而形成“排名下降”，扭曲分数。173因此，他们添加了一个额外的修正因子E，得到了以下修改后的方程，该方程（与前一个方程不同）的正式标题是“定义 1 ”：

There then follows some backtracking. While the authors maintain that it corresponds to “common sense,” this initial equation has a drawback: it is possible for groups of pages to mutually cite each other, which would form “rank sinks” that skew the scores.173 As a result, they add an additional corrective factor E, resulting in the following modified equation, which (unlike the foregoing one) is crowned with the formal heading “Definition 1”:

需要明确的是，这个等式本身并非一个算法，而是待计算函数的定义。这个定义并非Church意义上的递归，因为它是循环的：在展开R ′( u )的公式时，我们通常会回到原来的值。然而，这个等式可以通过构成PageRank算法的迭代方法求解，他们在题为“计算PageRank”的简短章节中通过几乎完全由符号组成的简洁伪代码描述了该算法。

This equation, it should be made clear, is not itself an algorithm, but rather a definition of the function to be computed. This definition is not recursive in Church’s sense because it is circular: in expanding the formula for R′(u), one will often come back around to that same value. The equation can, however, be solved through an iterative method that constitutes the PageRank algorithm, which they describe, in a brief section titled “Computing PageRank,” by means of a terse pseudocode consisting almost entirely of symbols.

这种证明算法合理性的方法，与莱布尼茨之后出现的符号方法中的口头定义有些相似。Page 及其同事回顾了 Jean Le Rond d'Alembert 对负数含义的详细讨论，谨慎地提供了一个概念性的他们为符号方程式寻找基础，甚至呼应弗朗西斯·马塞雷斯对“常识”的援引。然而，这种概念基础在符号出现后就消失了，这在 18 世纪是不可接受的。佩奇和他的合著者不仅根据数学考虑修改了直观合理的方程式，还对其进行了重新诠释。他们最初用引用计数来解释 PageRank，后来又根据人们在网上“冲浪”（浏览）的模拟提出了不同的基本原理。“冲浪者”通过随机点击链接从一个页面移动到另一个页面；有时冲浪者“感到无聊”并随机选择一个页面。174 PageRank相当于衡量这个冲浪者在大量试验中可能在每个页面上花费的时间。

This approach to justifying an algorithm bears some resemblance to the verbal definitions that came to attend symbolic methods in Leibniz’s wake. Recalling Jean Le Rond d’Alembert’s detailed discussion of what negative means, Page and colleagues are careful to provide a conceptual basis for their symbolic equation; they even echo Francis Maseres in their invocation of “common sense.” Yet, in a way that would have been unacceptable in the eighteenth century, this conceptual basis falls away once symbols arrive. Not only do Page and his coauthors revise the intuitively justified equation based on mathematical concerns; they also reinterpret it. Whereas they initially explain PageRank in terms of citation counting, they later present a different rationale based on a simulation of people “surfing” (browsing) the web. A “surfer” moves from page to page by randomly clicking links; occasionally the surfer “gets bored” and chooses a page at random.174 PageRank is equivalent to a measure of how much time this surfer is likely to spend at each page over a large number of trials.

这种可重新解释性极大地增强了经典算法的能力。在 Page 团队的白皮书发表几年后，Alon Altman 和 Moshe Tennenholtz 再次重新解释了 PageRank，这次将其作为投票过程的模拟。175该算法在生物化学领域找到了另一个应用，它可以用来分析蛋白质相互作用网络。176算法即使转移到完全不同的学科环境中，也可以保留稳定的逻辑核心，这种观点延续了形式规则优先于概念解释的思想，这种思想在 William Roman Hamilton 和 George Boole 时代改变了符号代数，而这隐含在 ALGOL 对形式语义和“语法糖”的区分中。这种基本原理被认为是逻辑的幌子，对算法的实际应用至关重要，但可以在方便的时候进行更换。

This reinterpretability contributes greatly to the power of classical algorithms. A few years after the Page group’s white paper was published, Alon Altman and Moshe Tennenholtz reinterpreted PageRank once more, this time as a simulation of a voting process.175 The algorithm found another application further afield in biochemistry, where it can be used to analyze protein interaction networks.176 The idea that algorithms can retain a stable logical core even as they are transferred to entirely different disciplinary contexts continues the privileging of formal rules over conceptual interpretations that transformed symbolic algebra in the time of William Roman Hamilton and George Boole, and that is implicit in ALGOL’s distinction between formal semantics and “syntactic sugar.” The rationale is figured as mere window dressing upon the logic, crucial to the practical application of algorithms but capable of being swapped out whenever it is convenient to do so.

这种方法的难点在于，它在算法教科书和研究论文中提出的技术定义（通常以方程式的形式）与用户的非技术视角之间造成了认识论上的鸿沟。PageRank 论文可能包含该方法的直观原理，但用户界面通常会跳过这一步。例如，当面对按“相关性”排序的搜索结果时，用户必须相信系统使用的任何计算都与相关性的含义相对应。177许多评论员表示担心，算法经常被隐藏起来，在 Facebook 和 Google 等公司的服务器场中运行，不受公众监督。经典范式中没有任何内容说算法必须隐藏；回顾《ACM 通讯》中的“算法”部分，发布算法是一种常见的做法，Quicksort、Dijkstra 算法和 PageRank 的解释都可以在维基百科上查阅。但将某些算法作为商业机密保留的做法只是更深层次分歧的一种表现。通过古典范式为机械操作提供了概念解释，它使程序员能够制定自己的技术定义，而无需更广泛社区的意见，而更广泛社区在普通的写作和演讲中对某个词的含义有一定的发言权。

A difficulty of this approach is that it creates an epistemological divide between the technical definitions presented (often in the form of equations) in algorithms textbooks and research papers and the nontechnical perspectives of users. The PageRank paper may include an intuitive rationale for the method, but user interfaces typically skip this step. When faced with search results sorted by “relevance,” for instance, the user has to trust that whatever calculation the system is using corresponds somehow to the meaning of relevance.177 Numerous commentators have expressed a concern that algorithms are often kept hidden, cranking away in the server farms of companies such as Facebook and Google, concealed from public scrutiny. There is nothing in the classical paradigm that says algorithms must be hidden; going back to the “Algorithms” section in Communications of the ACM, it has been a common practice to publish algorithms, and explanations of Quicksort, Dijkstra’s algorithm, and PageRank are up on Wikipedia for all to see. But the practice of keeping some algorithms as trade secrets is only one expression of a deeper divide. By subordinating conceptual explanations to mechanical operations, the classical paradigm enables programmers to enact their own technical definitions without the input of the broader community that, in ordinary writing and speech, has some say as to what a word means.

经典算法已被证明是一套可行的学科实践，即使其局限性很大。早期计算机科学家在“人为因素”与算法逻辑之间设置的障碍——计算内容的选择与高效计算的技术细节之间的障碍——使得一些方法取得了实际成果，其中包括用于解析Python和HTML等计算机语言语法的高度复杂程序。178 但计算在21世纪初开始脱离经典范式。在本世纪的前二十年，“算法”一词越来越多地指代机器学习（ML）系统，而非快速排序之类的人工设计程序。机器学习总是包含某种预先定义的目标，或至少是衡量成功的标准，但定义这一目标的方式却截然不同：在ML中，目标是通过数据而不是通过直接指定计算机要计算或逼近的数学函数来制定的。机器学习显然没有取代经典算法，但它在2010年代确实在计算系统中获得了新的突出地位。在本书的结尾，我思考了在二十一世纪的前二十年，随着机器学习在技术领域和公众对“算法”的想象中占据突出地位，关于算法的讨论发生了怎样的变化。

The classical algorithm has proven workable as a set of disciplinary practices, even if its limitations are substantial. The barrier that early computer scientists created between “human factors” and algorithmic logic—a barrier between the choice of what to compute and the technical details of how to compute it efficiently—enabled methods that have borne practical fruits, among them the highly complex procedures used to parse the syntax of computer languages like such as Python and HTML.178 But computation began to depart from the classical paradigm in the early twenty-first century. In the first two decades of the century, the word algorithm came increasingly to refer not to human-designed procedures such as Quicksort but to machine learning (ML) systems. Machine learning always involves some sort of predefined goal or at least measure of success, but the means of defining this goal is entirely different: in ML, the goal is laid out through data rather than through a direct specification of the mathematical function the computer is to compute or approximate. Machine learning has decidedly not replaced classical algorithms, but it did gain a newly prominent role in computational systems in the 2010s. In the coda to this book, I consider how the conversation about algorithms changed in the first two decades of the twenty-first century, as ML gained prominence both in technical fields and in the public imagination of what an “algorithm” is.

黑匣子和可调节旋钮

Black Boxes and Adjustable Knobs

如果说经典算法与人类理解相分离，那么它们也与数据相分离。1就快速排序的有效性而言，无论排序对象是什么，都无所谓——只要顺序定义明确，该程序对按播放时间排序的视频、按字母顺序排序的名称或按亮度排序的图像都同样有效。由于复杂性理论完全通过分析计算指令来运作，因此无需启动计算机即可严谨地研究经典算法。机器学习 (ML) 改变了这一点。“算法”不再由工程师设计，而是由机器根据大量数据进行调整。虽然这两种“算法”经常被一起讨论，但机器学习并非早期计算机科学家（如 Edsger Dijkstra 和 Donald E. Knuth）所创立的学科实践的直接延续。机器学习中“算法”的含义与其说源自计算机科学，不如说源自较新的数据科学学科，并且在两种语境中使用同一个词已经导致了很多混淆。 ML 与二十世纪提出的算法定义有所不同，这种差异对计算认识论具有广泛的影响。

If classical algorithms are divided from the human understanding, they are also divided from data.1 As far as the effectiveness of Quicksort goes, it makes no difference what one hopes to sort—provided the order is well defined, the procedure would work just as well with videos to be sorted by running time, names to be sorted alphabetically, or images to be sorted by brightness. Because complexity theory works solely through the analysis of computational instructions, it is possible to study a classical algorithm rigorously without ever even turning on a computer. Machine learning (ML) changes this. The “algorithms” are no longer designed by engineers but instead tuned by machines based on large amounts of data. While these two types of “algorithm” are often discussed together, ML is not a straightforward continuation of the disciplinary practices established by early computer scientists such as Edsger Dijkstra and Donald E. Knuth. The ML sense of algorithm emerged less from computer science than from the newer discipline of data science, and the use of the same word in both contexts has led to a great deal of confusion. ML departs from the definitions of algorithm put forth in the twentieth century, and the difference has wide-ranging implications for the epistemology of computation.

“机器学习”这一术语涵盖了范围广泛、复杂程度各异的技术。2 这一理念可以追溯到 20世纪40 年代，当时研究人员试图将人类认知结构形式化。3在自 20 世纪 50 年代以来，机器学习的研究开始与控制论领域（研究系统之间的动态关系）以及人工智能 (AI) 联系在一起。4尽管存在一些重叠，但这些研究项目与算法子领域基本不同；它们并非寻求精确定义的数学问题的解决方案，而是考虑机器与其环境的交互，并寻求自动化更开放的任务，例如翻译和模式识别。到 20 世纪 80 年代，现代机器学习的大部分技术设备已经到位，但这种实践在 21 世纪的第一个十年初期才开始显著发展，这在很大程度上归功于计算能力的提升以及非常大的数据集的易于获取。

The term machine learning can refer to a wide range of techniques that vary greatly in sophistication.2 The idea goes back to the 1940s, when researchers attempted to formalize the structures of human cognition.3 In the 1950s, this effort came to be associated with the field of cybernetics, which dealt with dynamic relationships between systems, and with artificial intelligence (AI).4 Although there was some overlap, these research programs were largely separate from the subfield of algorithms; rather than seeking solutions to precisely defined mathematical problems, they considered interactions of machines with their environments and sought to automate tasks that were more open-ended, such as translation and pattern recognition. Much of the technical apparatus of modern ML was already in place by the 1980s, but the practice gained greatly in prominence early in the first decade of the twenty-first century, owing in large part to an increase in computing power and in the easy availability of very large data sets.

为了说明这些技术中的一种，请考虑一个常见的应用：图像分类。要训练一个 ML 系统来区分马的图像和狗的图像，首先需要收集一组图像示例，所有图像示例都标注有正确的标签；这些标签有时被称为“黄金标准” ，这很有暗示性。还需要构建一个模型，即一个数学函数，其参数将在训练过程中确定。图像识别模型通常由人工神经网络 (ANN) 组成，它们的灵感来自人脑的结构，尽管它们最终只不过是由矩阵乘法序列和其他简单的数学函数组成的。接下来，必须指定一个损失函数来衡量输出的质量，数字越低越好。由于在这种情况下正确的标签是已知的，因此损失函数将衡量模型对训练数据的错误分类的严重程度。

As an example that illustrates just one of these techniques, consider a common application: image classification. To train an ML system to distinguish images of horses from images of dogs, one would first need to assemble a collection of image examples, all annotated with correct labels; these labels are sometimes called, suggestively enough, the gold standard. One would also need to construct a model, meaning a mathematical function whose parameters are to be determined in the training process. Image recognition models typically consist of artificial neural networks (ANNs), which are inspired by the structure of the human brain, although they ultimately consist of little more than sequences of matrix multiplications and other simple mathematical functions. Next, one must specify a loss function that measures the quality of the output, lower numbers being better. Since the correct labels are known in this case, the loss function would measure how badly the model misclassifies the training data.

一旦这些元素都组装好了，就可以开始训练了。通过梯度下降之类的优化程序，机器会使用微积分来确定每个参数的局部变化将如何影响输出，然后逐渐调整参数以减少损失。（顺便说一句，优化正是 1684 年 G.W. 莱布尼茨将微分方法称为“算法”时所要解决的问题，尽管方法并不完全相同。）训练结果是一个可以用来进行预测的训练模型。这种方法之所以令人兴奋，是因为它似乎可以处理以前超出计算机能力范围的任务。人类很难设计出明确的标准来判断一幅图像是否描绘了一匹马，但让机器根据大量图像示例“学习”该标准却非常有效（尽管它肯定不是完美的）。

Once these elements are assembled, one can begin the training. Through an optimization procedure such as gradient descent, which uses differential calculus to determine how local changes in each parameter would affect the output, the machine adjusts the parameters little by little so as to decrease the loss. (Optimization was, incidentally, just the problem G. W. Leibniz was addressing when he called the method of differentials an “algorithm” in 1684, although the approach is not quite the same.) The result is a trained model that can be used to make predictions. The excitement over this method stems from the fact that it can apparently handle tasks that had formerly been outside the scope of what computers could do. A human being would have a hard time devising explicit criteria for whether an image depicts a horse, but having a machine “learn” the criteria based on a large number of image examples works remarkably well (although it certainly does not work perfectly).

ML“算法”与算法教科书。训练好的ANN可以毫无问题地表达为明确定义的步骤——即矩阵乘法序列和其他数学运算。这些过程也与经典算法一样，独立于具体实施例；人们可以在ONNX和TorchScript等文件格式之间转换它们，同时保留其基本逻辑（尽管任何真正尝试过的人都可以证明，这种转换在实践中并非总是无损的）。更广泛地说，机器学习与经典范式一样，倾向于主要根据流程是否产生预期的输出来判断。理论上，机器学习系统的性能可以自动评估，而无需考虑其采用的操作是否对人类有意义。通过这种方式，机器学习将我认为是算法思维特征的主体与客体的分歧——正确性标准与人类思维形成之间的分歧——推向了极端。

ML “algorithms” have some similarities to the procedures cataloged in algorithms textbooks. Trained ANNs can, with no particular problem, be expressed as clearly defined steps—namely, sequences of matrix multiplications and other mathematical operations. These procedures also share classical algorithms’ independence from specific embodiments; one can convert them between file formats such as ONNX and TorchScript while preserving their fundamental logic (although, as anyone who has actually tried to do this can attest, the conversion is not always lossless in practice). More broadly, ML shares the classical paradigm’s tendency to judge processes primarily by whether they produce the desired output. An ML system’s performance can, in theory, be evaluated automatically without regard for whether the operations it employs make any sense to people. In this way, ML takes the subject–object divide that I have identified as characteristic of algorithmic thinking—a divide between standards of correctness and the formation of human thought—to an extreme.

然而，这些模型的内容与Knuth编写的问题求解程序截然不同，机器学习在计算逻辑与人类理解之间造成的差距日益凸显。经典算法通常采用“全有或全无”的规则：如果x的值是这个，就执行那个。一些机器学习模型，例如决策树，包含这样的条件规则，但ANN采用基于微积分的优化方法，而严格的“非此即彼”逻辑则与之格格不入。为了使梯度下降法发挥作用，每个操作都必须（有一些限制性例外）可微分，这大致意味着状态之间的转换必须遵循平滑曲线。5因此， ANN只能模糊地表示类别区别：一幅图像可能被归类为90%的狗和10%的马。诚然，这些模型必须简化为1和0才能在计算机上运行，​​但理论上它们适用于实数上的连续函数，而不是图灵机的离散符号。6

The content of these models, however, is rather different from the problem-solving procedures compiled by Knuth, and the gap ML can create between computational logic and human understanding has increasingly appeared as a problem. Classical algorithms usually employ all-or-nothing rules: if the value of x is this, do that. Some ML models, such as decision trees, contain such conditional rules, but ANNs employ calculus-based optimization methods to which strict either/or logic is inimical. For gradient descent to work, every operation must (with some restricted exceptions) be differentiable, roughly meaning that the transitions between states must follow smooth curves.5 As a result, ANNs can only represent categorical distinctions fuzzily: an image might be classified as 90 percent dog and 10 percent horse. It is true that the models must be reduced to ones and zeros to run on computers, but in theory they work with continuous functions over real numbers, not with the discrete symbols of Turing Machines.6

除了这些数学上的差异之外，机器学习还提出了一些认识论问题，这些问题与早期计算机科学家发展算法理论时提出的问题截然不同。在现实情况下，训练数据代表着更广泛的现象：一组照片可能非常大，但它无法涵盖马匹照片的所有形态。因此，即使模型在训练数据上取得了完美的结果，但在使用其他尚未见过的数据进行测试时，它仍然可能出错。这个问题的专业术语是泛化。7为了从数学上推理泛化，机器学习采用了理想化的假设，例如 iid（独立同分布），它断言所有数据都来自单一概率分布，并且不会相互影响。8但这种方法的有效性取决于计算中使用的数据评估指标。要证明一个图像分类器不仅能推广到测试数据，还能推广到任何数据，这需要莱布尼茨的普适性：一个完整​​且正确的万物理论。如果没有这样的理论，模型就仍然会存在争议，它能否达到CAR Hoare和John McCarthy所追求的可信度——这在自动驾驶汽车等生死攸关的应用中尤为重要——仍将备受争议。

Apart from these mathematical differences, ML raises distinct epistemological questions from the ones that led early computer scientists to develop a theory of algorithms. In realistic cases, the training data stand in for a broader range of phenomena: a collection of photographs may be very large, but it cannot encompass every form a photograph of a horse may take. As a result, even if the model gets perfect results on the training data, it can still go wrong when tested on other data that have not yet been seen. The technical term for this problem is generalization.7 To reason about generalization mathematically, ML employs idealizing assumptions such as i.i.d. (independent and identically distributed), which asserts that the data all come from a single probability distribution and do not affect each other.8 But this approach is only as good as the data used in computing evaluation metrics. Proving that an image classifier can generalize not just to the test data but to any data whatsoever would require nothing short of Leibniz’s universal characteristic: a complete and correct theory of everything. Absent such a theory, a model will remain open to dispute, and whether it can ever attain the level of trustworthiness C. A. R. Hoare and John McCarthy sought—an issue especially important in life-and-death applications such as autonomous cars—will be debatable.

机器学习的兴起恰逢“算法”一词用法的另一次转变。无论是在流行话语中还是在技术领域，算法现在有时都用于指代 ML 系统，而不是快速排序和 A* 等分步过程。这种转变部分源于该词在统计学领域的采用，该领域的优先级与软件工程不同。在 2001 年的一篇题为“统计建模：两种文化”的文章中，Leo Breiman 对比了两种他称之为“数据建模”和“算法建模”的思想流派。9 他声称数据建模方法在学术界占主导地位，这种方法试图通过建立与可理解的理论相对应的模型来解释数据。他声称算法建模在商业中更为常见，其目的仅是进行预测，将模型视为“黑匣子”，仅通过其输出的准确性来判断。10 Breiman 认为，学术界忽视了黑箱模型的潜力，为了所谓的“无关理论”而牺牲了预测的准确性。11在Breiman 的影响下，“算法”一词与计算机科学教科书中编写的抽象程序的联系越来越少，而更多地与以结果为导向的统计数据处理联系在一起。

The rise of machine learning coincided with yet another shift in the usage of the word algorithm. Both in the popular discourse and in technical fields, algorithm is now sometimes used to refer to ML systems rather than to step-by-step procedures such as Quicksort and A*. The shift stems in part from the word’s adoption in a field with disparate priorities from those of software engineering: statistics. In a 2001 article titled “Statistical Modeling: The Two Cultures,” Leo Breiman contrasts two schools of thought that he calls “data modeling” and “algorithmic modeling.”9 The data modeling approach, which he claims is dominant in academia, attempts to explain the data by producing a model that corresponds to a comprehensible theory. Algorithmic modeling, which he claims is more common in business, aims only to make predictions, treating the model as a “black box” to be judged solely by the accuracy of its output.10 Breiman argues that academics have overlooked the potential of black box models, sacrificing predictive accuracy for the sake of what he calls “irrelevant theory.”11 In Breiman’s wake, the word algorithm came to be associated less with the abstract procedures compiled in computer science textbooks and more with a results-oriented flavor of statistical data crunching.

Breiman 的文章以两种不同的方式使用“算法”这个词，这两种方式都与 Knuth 和其他早期计算机科学家的使用方式不完全对应。有时，他用它来指 Bernhard Rieder 所说的“算法技术”：设计计算系统的通用方法。12例如，Breiman 将“神经网络和决策树”称为“用于拟合数据的强大新算法”。13 Breiman声名鹊起是因为发明了这样一种技术：随机森林。14从这个意义上讲，算法是用于构建和训练模型的一组技术，而不是模型本身。然而，Breiman的“统计建模”文章包含一些表明并非如此的用法。他写道，算法建模的目标是“找到一个算法f ( x )，使得对于测试集中的未来x ， f ( x ) 将是y的良好预测器”。15 在这种情况下，算法似乎是通过训练过程找到的预测函数。正是在第二种意义上，人们谈论识别马匹或预测贷款违约的算法：算法不是一种通用技术，而是计算密集型数据处理过程的特定结果。

Breiman’s article uses the word algorithm in two distinct ways, neither of which exactly corresponds to how it is used by Knuth and other early computer scientists. At times, he uses it to refer to what Bernhard Rieder has called “algorithmic techniques”: general methods for designing computational systems.12 Breiman refers, for instance, to “neural nets and decision trees” as “powerful new algorithms for fitting data.”13 Breiman’s claim to fame is inventing one such technique: random forests.14 In this sense, an algorithm is a set of techniques used to construct and train a model, not the model itself. Yet Breiman’s “Statistical Modeling” article contains some usages that suggest otherwise. The goal of algorithmic modeling, he writes, is “to find an algorithm f (x) such that for future x in a test set, f (x) will be a good predictor of y.”15 The algorithm, in this case, would seem to be the predictive function found through the training process. It is in this second sense that one talks about algorithms recognizing horses or predicting loan defaults: the algorithm is not a general technique but the particular result of a computationally intensive process of data crunching.

当布雷曼试图捍卫自己的立场时，这两种“算法”含义的共存导致了误解。这篇名为《统计建模》的文章与一系列回应和反驳一同发表。其中一位回应者，斯坦福大学的统计学家布拉德利·埃夫隆指出，这篇文章乍一看“像是在反对简约性和科学洞察力”，并指出：“新算法通常以黑匣子的形式出现，带有大量可调参数（‘旋钮’）。”布雷曼回复道：

The coexistence of these two senses of algorithm led to a misunderstanding when Breiman moved to defend his position. The “Statistical Modeling” article was published along with a collection of responses and rebuttals. One of the respondents, the Stanford statistician Bradley Efron, notes that the article looks at first glance “like an argument against parsimony and scientific insight” and states: “The new algorithms often appear in the form of black boxes with enormous numbers of adjustable parameters (‘knobs to twiddle’).”16 Breiman replies:

这句话有点让人困惑，也许我没理解Brad的意思。随机森林每次运行只需设置一个可调参数，并且对这个参数在很大范围内的值不敏感，而且可以快速简单地确定一个合适的值。支持向量机则依赖于1-2个参数的设置。其他算法模型在需要调整的参数数量上也同样稀疏。17

This is a perplexing statement and perhaps I don’t understand what Brad means. Random forests has only one adjustable parameter that needs to be set for a run, is insensitive to the value of this parameter over a wide range, and has a quick and simple way for determining a good value. Support vector machines depend on the settings of 1–2 parameters. Other algorithmic models are similarly sparse in the number of knobs that have to be twiddled.17

分歧似乎在于，“算法”指的是通过训练过程设定参数的函数（Efron），还是指生成该函数的方法（Breiman，至少在本例中如此）。Efron 认为，由于偏差和方差之间的权衡，参数过多的模型会导致有偏差的估计。从这个角度来看，做出选择的人还是机器并没有什么区别。Breiman 更重视机器/人类（或至少是算法/非算法）的区别。机器学习模型可能拥有大量的学习参数（据报道，GPT-3 拥有 1750 亿个参数），但这些参数是在黑匣子内确定的，因此 Breiman 并不认为它们是“可调整的”。

The disagreement, it seems, comes down to whether “algorithm” refers to the function whose parameters are set through the training process (Efron) or to the method used to produce this function (Breiman, at least in this instance). Efron argues that, because of the trade-off between bias and variance, models with too many parameters will lead to biased estimates. In this regard, it makes little difference whether a human or a machine is making the choices. Breiman gives much more significance to the machine/human (or at least algorithmic/nonalgorithmic) distinction. An ML model may have an enormous number of learned parameters (GPT-3 reportedly has 175 billion), but these parameters are determined within the black box and so Breiman does not count them as “adjustable.”

Efron 和 Breiman 之间的争论不仅仅是术语问题。正如我在第五章关于 PageRank 的讨论中所指出的，将计算方法应用于诸如网页排名之类的定义模糊的问题，可能会引发对任意性的担忧——一种人们可以通过调整设置来得到任何想要的结果的感觉。PageRank 的作者们用概念性的理由来应对这种担忧，认为他们的定义符合“常识”。同样，Efron 也从传统的立场出发，认为统计模型应该有理论基础。Breiman 的文章显然主张实用性而非理论化，但它也提供了一种缓解对任意性的担忧的方法，即专注于通过实证试验衡量的预测准确性。这种关注导致（在 Kaggle 等网站上）出现了一种游戏化的研究形式，在这种研究中，研究者们并非在证明其方法的合理性。用 PageRank 论文的修辞方式来说，数据科学家们争夺基于指标的排行榜的顶级位置。18

The dispute between Efron and Breiman is not merely a matter of terminology. As I argued in the discussion of PageRank in chapter 5, applying computational methods to loosely defined problems such as ranking web pages can inspire an anxiety about arbitrariness—a sense that people can coax out whatever results they want by tweaking settings. The PageRank authors address this anxiety with a conceptual rationale, arguing that their definition corresponds to “common sense.” Similarly, Efron writes from the traditional position that statistical models should have theoretical bases. Breiman’s article is overtly an argument for practicality over such theorizing, but it also provides a way of assuaging the anxiety about arbitrariness by placing the exclusive focus on predictive accuracy as measured through empirical trials. This focus has led (on websites like Kaggle) to a gamified form of research in which, rather than justifying their methods rhetorically in the manner of the PageRank paper, data scientists vie for top positions on metric-based leaderboards.18

虽然机器学习文献中“算法”一词的用法各不相同，但对于机器学习系统的哪一部分构成“算法”，人们普遍存在模糊定义。例如，Ian Goodfellow、Yoshua Bengio 和 Aaron Courville 在 2016 年出版的教科书《深度学习》给出了以下非正式定义：“机器学习算法是一种能够从数据中学习的算法。” 19在教材的后面，他们给出了构建深度学习算法的更具体的“秘诀”：“结合数据集规范、成本函数、优化程序和模型。” 20这个秘诀表明，“机器学习算法”是一组可重复使用的技术，用于构建和训练模型，这主要是作者使用该术语的方式：例如，他们将线性回归称为“一种简单的机器学习算法” 。21

While the usage of the term algorithm in the ML literature varies, the equivocation as to which part of an ML system constitutes the “algorithm” has become widespread. The 2016 textbook Deep Learning by Ian Goodfellow, Yoshua Bengio, and Aaron Courville, for instance, gives the following informal definition: “A machine learning algorithm is an algorithm that is able to learn from data.”19 Later in the text, they give a more specific “recipe” for the construction of deep learning algorithms: “combine a specification of a dataset, a cost function, an optimization procedure and a model.”20 This recipe suggests that a “machine learning algorithm” is a reusable set of techniques used to construct and train models, which is mainly how the authors use the term: they refer, for instance, to linear regression as “a simple machine learning algorithm.”21

然而，算法“能够学习”的说法与这一概念并不相符——它至少模棱两可地表明，算法会随着时间而变化，在梳理数据时扩展其能力。这一命题只有当“算法”不仅包含构建机器学习系统所使用的技术，还包含机器在训练过程中调整的参数值时才有意义。作者使用“学习算法”而不是“训练算法”来强化这种自主性。这种歧义在非正式讨论中更为普遍，人们通常说“算法”拥有一系列后天习得的能力，从预测欺诈到识别人脸再到翻译文本。算法之间界限的模糊可能因新的开发工具而得到加强，例如 Jupyter Notebook，它使程序员能够在程序变量加载到内存中时编辑程序；因此，训练过程和学习到的参数看起来像是一个动态变化系统的一部分。这种“算法”，如果这个概念真的是连贯的，那么它与快速排序是非常不同的，而马尔可夫和克努斯提出的定义对于解释它的价值有限。22

The statement that algorithms are “able to learn,” however, sits uneasily with this conception—it suggests, at least equivocally, that the algorithm changes over time, expanding its capacities as it combs through data. This proposition only makes sense if the “algorithm” encompasses not just the techniques used in constructing an ML system but also the parameter values as they are adjusted by the machine during training. The authors reinforce this assignment of agency by their use of the phrase learning algorithm rather than training algorithm. The ambiguity is far more prevalent in informal discussions, where “algorithms” are commonly said to possess a range of acquired abilities, from predicting fraud to identifying faces to translating text. The blurring of the lines around algorithms has likely been reinforced by new development tools such as Jupyter Notebook, which enables programmers to edit a program while its variables are loaded in memory; training procedures and learned parameters can thus seem like parts of a single dynamically changing system. This type of “algorithm,” if the concept really is a coherent one, is a very different beast from Quicksort, and the definitions put forth by Markov and Knuth are of limited value in explaining it.22

这种语义上的转变确实值得惋惜。将算法与机器学习联系起来，并不能很好地描述算法课程中涵盖的那些定义明确的程序。这些程序虽然不如机器学习那么浮夸，但对编程学科来说，现在和将来都至关重要。事实上，经典算法在机器学习本身中也发挥着作用：《深度学习》教科书包含许多人为设计的用于训练和推理的程序，并以“算法 6.3”之类的标题开头。23本书以伪代码的形式呈现这些程序，基本遵循了经典范式，但值得注意的是，作者通常不指定所需的输出。“机器学习算法”究竟是指这些人工设计的训练程序，还是指经过训练的预测模型，这种模糊性导致了大量讨论和混淆。在这些方法被广泛炒作的背景下，“算法”一词被赋予了一系列含义，这些含义更多地与市场营销相关，而非与严肃的技术分析相关，评论员最好对“算法”一词的不精确使用持怀疑态度。24

There is some reason to lament this semantic shift. Associating algorithm specifically with machine learning leaves behind no good term for the precisely defined procedures covered in algorithms courses, which, while less flashy than ML, are and will remain important to the discipline of programming. Indeed, classical algorithms play a role in ML itself: the Deep Learning textbook includes numerous human-designed procedures for use in training and inference, set off with such headings as “Algorithm 6.3.”23 The book presents these procedures in a pseudocode form that basically follows the classical paradigm, although it is notable that the authors usually do not specify the desired output. The ambiguity as to whether “machine learning algorithm” refers to these human-designed training procedures or to trained predictive models has led to a great deal of cross talk and confusion. Amid the widespread hype about these methods, the word algorithm has taken on a range of connotations more aligned with marketing than with serious analyses of technology, and commentators would do well to meet imprecise uses of the term with skepticism.24

然而，对计算的思考方式的更广泛转变可能是一个积极的发展。经典范式将社会问题抽象化，将其包含在正式的问题定义中，从而鼓励人们狭隘地关注技术细节，而忽略了系统应用的更广阔背景。机器学习研究也可能持有类似的狭隘观点，因为它将目光投向了固定的基准；然而，机器学习方法的实际应用使得社会“外部性”更难以忽视。毕竟，总得有人设定目标，而且，由于缺乏强有力的论证，模型、性能指标和数据集设计中的众多决策为任意性卷土重来创造了机会。因此，机器学习与身份政治如此激烈地交锋也就不足为奇了：对任意性的焦虑自然会促使人们质疑，究竟是谁在仲裁，又是谁在操控着决定机器学习模型运行的旋钮。无论如何，机器学习正在打破学科界限，并为我们提供了一个重新思考计算与人类思维和语言之间关系的基本假设的机会。GPT-3 就是一个很好的例子。

Yet the broader shift in thinking about computation could be a positive development. The classical paradigm abstracts out social issues by containing them in formal problem definitions, thus encouraging a narrow focus on technicalities to the exclusion of the broader contexts in which systems will be employed. Machine learning research can take a similarly blinkered view to the extent that it sets its sights on fixed benchmarks; yet the practical adoption of ML methods has made social “externalities” harder to dismiss. Someone, after all, must set the goals, and, in the absence of strong rhetorical justifications, the numerous decisions that go into the design of models, performance metrics, and data sets create opportunities for arbitrariness to slip back in. It is not surprising, as a result, that ML has reacted so explosively with identity politics: the anxiety about arbitrariness naturally leads one to question who is arbitrating, who is turning the knobs that determine what ML models do. For better or worse, ML is breaking down disciplinary boundaries, and it opens an opportunity to rethink, once again, fundamental assumptions about how computation relates to human thought and language. A case in point is GPT-3.

机器学习与生活语言

Machine Learning and the Living Language

与我在第 5 章讨论的算法不同，GPT-3 在很大程度上是黑盒化的。截至撰写本文时，经过训练的模型尚未向公众开放；它运行在服务器上，只有获得许可才能与之交互。然而，GPT-2 可以下载，而且正如研究论文中所述，关于这些技术背后的通用方法有很多值得探讨的地方。作为“多任务学习器”，GPT 模型应该适用于一系列领域，但不是通过我们在 PageRank 中看到的那种概念重新诠释来实现的。相反，我们应该通过输入用英语书写的输入来诱导程序执行任务——例如，“ What is the capital of Belgium?”或“ Rewrite the following passage in the style of Herman Melville.”。这些提示中的单词并非像 ALGOL 中的标识符那样“本质上毫无意义”，而是必须以与模型所基于的人类书写中的用法相对应的方式使用。训练有素。用自然语言与计算机交流的想法并不新鲜；它至少可以追溯到 Jean E. Sammet 1965 年关于英语编程的演讲。然而，这种可能性直到 2010 年代才接近实现，而这种新的技术幻想有望彻底改变机器与人类的互动方式。通过赋予数学和语言同等的角色，GPT-3 将自乔治·布尔时代以来一直存在于符号方法中的主客体二元论推向了临界点，这种二元论构成了经典算法概念和现代编程语言的基础。

Unlike the algorithms I discussed in chapter 5, GPT-3 is black-boxed in a strong sense. The trained model, as of this writing, is inaccessible to the public; it runs on servers with which one can interface only by permission. It is, however, possible to download GPT-2, and there is much to be said about the general methods behind these technologies, as described in research papers. As “multitask learners,” the GPT models are supposed to be applicable in a range of domains, but not through the sort of conceptual reinterpretation we saw with PageRank. Instead, one is supposed to induce the program to perform tasks by feeding in inputs written in the English language—for example, “What is the capital of Belgium?” or “Rewrite the following passage in the style of Herman Melville.” The words in these prompts are not “inherently meaningless,” as identifiers in ALGOL are supposed to be, but instead must be used in ways that correspond to their usage in the human writings on which the model is trained. The idea of communicating with a computer in natural language is not new; it goes back at least to Jean E. Sammet’s 1965 talk about programming in English. Yet the possibility did not come close to realization until the 2010s, and this new technological chimera promises a radical shift in how machine and human interact. By granting mathematics and language coequal roles, GPT-3 pushes the subject–object dualism that has attended symbolic methods since the time of George Boole, and that forms the basis of both the classical conception of algorithm and the modern programming language, to a breaking point.

GPT 模型基于一种称为 Transformer 的 ANN 架构，该架构于 2017 年首次推出。25 Transformer是语言模型，即经过训练可以预测在特定语境中会出现什么单词的模型。GPT 模型会预测给定单词序列之后的下一个单词（例如，I bought some food at the grocery →预测store：），而其他模型，例如由 Jacob Devlin 领导的 Google AI 研究团队开发的 BERT（来自 Transformers 的双向编码器表示），则经过训练可以填空，类似 Mad Libs 风格。26 Transformer架构包含两部分：编码器和解码器。完整的 Transformer（例如 T5）应该能够通过将文本编码为数字形式，然后用目标语言解码，从一种语言翻译成另一种语言。27 GPT模型主要用于文本生成，仅由解码器组成。通过更换不同的“头部”（在神经网络末端添加的额外层），可以将 Transformer 模型应用于一系列任务，利用其在训练过程中学到的知识，实现诸如回答问题、查找信息、文本分类和自动摘要等功能。此外，还可以通过输入专门设计的输入（至少在理想情况下）来诱导生成器生成所需的输出，将文本生成头部用于其他任务。

The GPT models are based on an ANN architecture called the Transformer, which was first introduced in 2017.25 Transformers are language models, meaning models trained to predict what word will appear in a certain context. The GPT models predict the next word after a given sequence of words (e.g., I bought some food at the grocery → prediction: store), whereas other models such as BERT (Bidirectional Encoder Representations from Transformers), which was developed by a Google AI Research team led by Jacob Devlin, are trained to fill in the blanks, Mad Libs style.26 The Transformer architecture has two parts, an encoder and a decoder. Full Transformers such as T5 are supposed to be able to translate from one language to another by encoding the text into a numerical form and then decoding it in the target language.27 The GPT models, which are primarily intended for text generation, consist only of decoders. By swapping in different “heads”—extra layers added to the end of the neural network—it is possible to apply Transformer models to a range of tasks, leveraging whatever they learned in the training process for such purposes as answering questions, finding information, classifying texts, and performing automatic summarization. It is also possible to use text generation heads for other tasks by feeding in specially designed inputs that (at least ideally) induce the generator to produce the desired output.

语言模型的概念起源于 20 世纪 C.E. 香农 (C.E. Shannon) 和沃伦·韦弗 (Warren Weaver) 的研究成果。在他 1948 年关于信息论的开创性论文中，香农描述了“ n - gram”模型，该模型（基于 A.A. 马尔可夫 Sr. 的前期工作）确定哪些单词或字符最有可能出现在特定固定长度序列之后。28因此，在“ ENGLISH ”之后，香农的模型猜测下一个单词将是“ WRITER ”。29这些模型最初的目的是描述语言的统计特性，以便更好地解释交流信道中的噪声。在论文的第二段，香农进行了一次臭名昭著的学科边界设定：“信息通常具有意义；也就是说，它们指的是某些系统，或者根据某些物理或概念实体与之相关。这些语义方面交流与工程问题无关。” 30 Weaver 在 1949 年与香农论文同时出版的一本书中的配套文章中详细阐述了这一点，区分了交流理论的“三个层次”。31 “技术层次”与符号是否准确传输有关，“语义层次”与这些符号的含义有关，“有效性层次”与含义的选择是否恰当有关。根据 Weaver 的说法，语言模型只处理技术层面，尽管他也指出，其他两个层次“有很多东西可以借鉴”语言的统计分析。32

The idea of a language model originated in the twentieth-century work of C. E. Shannon and Warren Weaver. In his germinal 1948 paper on information theory, Shannon described the “n-gram” model, which (building on the prior work of A. A. Markov Sr.) determines which words or characters are most likely to appear after certain sequences of a fixed length.28 Thus, after “ENGLISH,” Shannon’s model guesses that the next word will be “WRITER.”29 The original purpose of these models was to describe statistical properties of language so as to better account for noise in communication channels. In the second paragraph of the paper, Shannon performs a notorious act of disciplinary boundary setting: “Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem.”30 Weaver expands on this point in a companion essay that was published together with Shannon’s paper in a 1949 book, distinguishing “three levels” in the theory of communication.31 The “technical level” has to do with whether symbols are transmitted accurately, the “semantic level” with what those symbols mean, and the “effectiveness level” with whether that meaning is chosen appropriately. According to Weaver, language models deal only with the technical level, although he also notes that the other two levels “have much to learn from” a statistical analysis of language.32

GPT-3 的设计者将语言模型呈现为多任务学习器，从而采纳了 Weaver 让统计数据与语义对话的想法。然而，在其他方面，GPT-3 又背离了 20 世纪的学科范畴。除了与香农合作之外，Weaver 还是机器翻译的早期倡导者，并在 1949 年的一份备忘录中对此进行了讨论。在这份备忘录中，Weaver 认为所有语言都有一个共同的逻辑核心。他认为，跨语言翻译的最佳方式“是从每种语言下降到人类交流的共同基础——真实但尚未被发现的通用语言——然后通过任何方便的特定途径重新出现。” 33这种说法呼应了约翰·威尔金斯的真实性格和 G.W. 莱布尼茨的普遍性背后的普遍主义思想，假设某些概念集为所有语言所共有。但与 ALGOL 一样，Weaver 的普遍性也被一条将逻辑与文化区分开来的学科界限所阻隔。韦弗认为，机械方法可能更难应用于“文学翻译”，因为在文学翻译中，语言的“非逻辑”元素，如“直觉的风格、情感内容等”至关重要。34韦弗在这里划的界线很像布尔在《思维定律》最后一章中设立的界线。一边是词语的字面意义及其逻辑关系，另一边是感觉、美学和特定语言的独特品质——简而言之，所有通常被视为人文学科的事情。

By presenting language models as multitask learners, the designers of GPT-3 take up Weaver’s idea of making statistics speak to semantics. In other ways, however, GPT-3 departs from twentieth-century disciplinary categories. In addition to his work with Shannon, Weaver was an early advocate of machine translation, which he discussed in a 1949 memo. In this memo, Weaver argues that there is a logical core common to all languages. The best way to translate between languages, he argues, “is to descend, from each language, down to the common base of human communication—the real but as yet undiscovered universal language—and then re-emerge by whatever particular route is convenient.”33 This claim echoes the universalist thinking behind John Wilkins’s real character and G. W. Leibniz’s universal characteristic, presuming that a certain set of concepts is common to all languages. But as with ALGOL, Weaver’s universality is cut across by a disciplinary line dividing logic from culture. Mechanical methods, Weaver states, may be harder to apply to “literary translation,” in which “alogical” elements of language such as “intuitive sense of style, emotional content, etc.” are of crucial significance.34 Weaver here draws a line much like the one Boole set up in the last chapter of The Laws of Thought. On the one side are the literal meanings of words and their logical relations, and on the other are feeling, aesthetics, and the distinctive qualities of particular languages—in short, all that would typically be viewed as matters for the humanities.

经验表明，韦弗对机器能力的判断是错误的：机器最擅长操控的正是他所谓的语言“非逻辑”方面。GPT 模型已被证明能够实现“风格迁移”，即用不同作者的风格重写一段文字。而生成逻辑连贯的文本则变得更加困难。不妨看看下面这段机器生成的段落，它出自介绍 GPT-2 的博客文章：

Experience has shown that Weaver was wrong about what machines could do: it is precisely what he called the “alogical” aspects of language that machines are best able to manipulate. The GPT models have proven capable of “style transfer,” meaning rewriting a passage in the style of a different writer. Generating logically coherent text has turned out to be more difficult. Consider this machine-generated paragraph, from the blog post introducing GPT-2:

找出内战爆发的原因很容易，因为许多人、许多书籍、许多电视和电影都告诉我们，原因，与种族、经济或宗教有关。人们普遍认为，这本质上是一场为了资本主义而进行的奴隶制战争，一场关于一个世纪奴隶制的战争。但大多数人想到内战时并非如此。许多人认为这场战争关乎州权。你说得没错，但说内战是州权与联邦制之间的冲突，这有点误导。我们再试一次。是什么激励了这么多人走上葛底斯堡、安提塔姆、葛底斯堡、彼得堡和弗雷德里克斯堡的战场？美国的共和理念——有限政府的理念——是历史的重要组成部分。35

It is easy to identify why the Civil War happened, because so many people and so many books and so much television and films tell us that it was the cause, that it has something to do with race or economics or religion. There is much agreement that it was essentially a war of slavery on behalf of capitalism, about a century of slavery. But that’s not what most people think of when they think of the Civil War. Many people think of the war as about states rights. You’re not wrong about that, but it’s kind of misleading to say that the Civil War was a conflict between states’ rights and federalism. So let’s try again. What’s the idea that has inspired so many people to go onto the battlefields of Gettysburg and Antietam and Gettysberg and Petersburg and Fredericksburg? The American idea of the republic—a notion of limited government—is a great part of the history.35

这类文本与14世纪的奥卡姆的威廉（William of Ockham）到1980年代的埃兹格·迪杰斯特拉（Edsger Dijkstra）等思想家所设想的那种全是实质内容、没有风格的逻辑语言截然相反。如果用一个非常古老的比喻来说，理解文本就像消化，那么阅读机器生成的文本有时就像闻到食物的味道却什么也没吃：人们能感觉到有人在说什么，但却永远无法确切地说出那是什么。

Such text is the polar opposite of the all-substance, no-style logical language imagined by thinkers from William of Ockham in the 1300s to Edsger Dijkstra in the 1980s. If understanding a text is, to use a very old analogy, like digestion, then reading machine-generated text can at times seem like smelling food without eating anything: one gets the sense that something is being said, but one can never pin down exactly what that something is.

这并不是说开发人员放弃了逻辑。更准确地说，他们颠倒了这些术语：逻辑不是计算方法的起点，而是目标。GPT-3 的技术论文将模型执行三位数算术的能力作为系统涌现智能的标志。因此，一个几乎显而易见的转折是，“算法”应该能够自行学习如何执行最初被称为“算法”的程序。其目标是创建一个能够执行一系列逻辑任务（例如“常识推理”）的系统。36

This is not to say that the developers have abandoned logic. It would be more accurate to say that they have reversed the terms: logic is not the starting point for computational methods but the goal. The technical paper about GPT-3 presents the ability of the model to perform three-digit arithmetic as a sign of the emergent intelligence of the system. Thus, in a twist that is almost too obvious to point out, an “algorithm” is supposed to learn on its own how to perform the procedures that originally went by the name algorithm. The aim is to produce a system that is capable of a range of logical tasks such as “commonsense reasoning.”36

与传统范式不同，无法用数学确定性来判断软件是否正确执行了这些任务。相反，研究人员采用诸如 SQuAD（斯坦福问答数据集）之类的基准，它们与标准化测试非常相似。37 SQuAD由一系列短文组成，并附有需要使用文章中的引文来回答的问题。例如，“什么？”“由市场和经济控制[原文如此]？”正确答案：“工人[原文如此]的工资。” 38问答任务有多个版本；有些是在提供的文章中找到答案，而另一些则进行“闭卷”测验，其中信息必须在模型本身中找到。回答问题的能力应该表明语言模型可以“理解”文本，从而执行它们在 Google 搜索和 Alexa 等系统中所使用的那种信息提取任务。

Unlike in the classical paradigm, there is no way to judge with mathematical certainty whether the software is performing these tasks correctly. Instead, researchers employ benchmarks such as SQuAD (Stanford Question Answering Dataset), which resemble nothing so much as standardized tests.37 SQuAD consists of a series of short passages accompanied by questions to be answered using quotations from the passages. “What,” for instance, “is controled [sic] by the market and economy?” Correct answer: “workers [sic] wages.”38 There are multiple versions of the question-answering task; some are about finding the answer in a supplied passage, whereas others administer a “closed-book” quiz, in which the information must be found in the model itself. The ability to answer questions is supposed to demonstrate that language models can “understand” texts and thus perform the sorts of information extraction tasks for which they are used in systems such as Google search and Alexa.

关于这样的系统是否真的能够“思考”，已经有很多文章进行了探讨，这个问题触及心灵哲学的核心问题。更重要的是，这些模型与语言的关系。首先，GPT-3 绝不是定义连贯的“英语”的直接表征。用于生成 GPT 模型的训练数据主要包括来自互联网的书籍和文本，这些文本通过半自动化方式过滤，以消除难以理解或其他不良文本。定义质量标准并非易事，而且这些数据集也不符合语言学家构建语料库时遵循的标准。由于很大一部分文本来自互联网讨论网站，这些模型倾向于重现那些在这些网站上花费最多时间的人的观点。训练数据还包括各种方言和克里奥尔语的文本，以及来自不同时期的文本——例如，这些模型显然接触过莎士比亚的作品——但这些文本并不一定附带任何关于其创作地点和时间的信息。英语是一种活的语言，以多种形式存在并随着时间而变化，模型只能解释这些差异，因为关于它们的信息可以以某种方式从单词本身的模式中抽象出来。

Much has been written about whether such a system can really “think,” a question that touches on central problems in the philosophy of mind. More to the present point is how the models relate to language. To begin with, GPT-3 is in no way a straightforward representation of a coherently defined “English language.” The training data that were used to produce the GPT models consist primarily of books and text from the internet, filtered through semiautomated means to eliminate unintelligible or otherwise undesirable text. Defining the criterion of quality is a troublesome matter, and the data sets fall short of the standards linguists follow in constructing corpora. Since a large part of the text comes from internet discussion sites, the models tend to reproduce the viewpoints of the people who spend the most time on those sites. The training data also include text in various dialects and creoles, as well as from a range of time periods—the models have, for instance, clearly been exposed to Shakespeare—but the texts are not necessarily accompanied by any information about where or when they were written. English is a living language, existing in many forms and changing over time, and the models can only account for these differences insofar as information about them can in some way be abstracted from patterns in the words themselves.

相关的观点是，语言模型依赖于语言标准化的历史进程，或者至少受益于这一进程。从17世纪开始，词典和语法书鼓励书面英语的规律性，从而导致了拼写的标准化。在此之前的几个世纪里，没有一种英语方言占据主导地位，拼写混乱：谚语中的抄写员可能会在同一页上用三种不同的方式拼写同一个单词。英语发展早期阶段的现存文本远远不足以训练出像GPT-3这样规模的语言模型，因此我们只能推测它的效果如何。39无论如何，任务都会有所不同。该模型无法假设字母序列和抽象意义上的“单词”之间存在稳定的对应关系，当不同地区和作者的用法差异很大时，也很难确定语法形式的语义。如果语言模型是通过抽象来运作的，那么其中一些抽象已经通过英语语言本身的标准化完成了。

A related point is that language models depend on, or at least benefit from, a historical process of language standardization. Starting in the seventeenth century, dictionaries and grammar books encouraged regularity in written English, leading to the standardization of spelling. For centuries before then, no single dialect of English was clearly dominant, and spelling was chaotic: the proverbial scribe might spell the same word three different ways on the same page. There is not nearly enough surviving text from this earlier stage in the development of English to train a language model on the scale of GPT-3, so we can only speculate as to how well it would work.39 The task would, in any case, be different. The model would not be able to presume a stable correspondence between sequences of letters and “words” in the abstract sense, nor would it be so easy to pin down the semantics of grammatical forms when usages differ widely between regions and writers. If language models work through abstraction, some of that abstraction has already been done through the standardization of the English language itself.

且不论所表征的语言类型，语言模型与语言之间存在着更广泛的脱节，这源于 ANN 的工作方式。训练过程完成后，ANN 是静态的；当 GPT-3 用于生成文本或执行其他任务时，它不会在此过程中“学习”任何新知识。批评人士指出，这种静态模型可能会导致“价值锁定”，即一项技术僵化。在其设计时就存在的实践。40虽然这种特定的训练范式可能会发生变化，但关于如何确定模型的成功仍然存在一个更广泛的问题。因为 ML 模型在输入未见过的输入时无法满足任何关于输出应该是什么的精确定义的标准，所以如果不借助某种形式的语言知识，就无法判断文本生成器对语言的模仿是否足够好。这种知识可能包括母语人士的水平，在这种情况下，评估模型则是调查和焦点小组的问题。如果涉及学科，那么我们必须转向古老的语法、修辞和逻辑三学科，以及文学美学，以及与 Dijkstra 划分学科界限的方式最相反的解释学。

Setting aside the issue of what language is being represented, there is a broader disconnect between language models and language that stems from how ANNs work. After the training process is complete, the ANN is static; when GPT-3 is used to generate text or perform other tasks, it does not “learn” anything new in the process. Critics have noted the possibility that such static models will lead to “value-lock,” in which a technology ossifies practices that existed at the time of its design.40 While this specific training paradigm is liable to change, a broader issue remains regarding how the model’s success is determined. Because ML models do not answer to any precisely defined standard of what the output should be when they are fed unseen input, there is no way of judging whether a text generator’s imitation of language is good enough without resorting to some form of linguistic knowledge. This knowledge could consist of the competency of native speakers, in which case evaluating the model is a matter for surveys and focus groups. If academic disciplines are at issue, then we must turn to the ancient trivium of grammar, rhetoric, and logic, along with literary aesthetics and, most antithetically to Dijkstra’s way of drawing the disciplinary lines, hermeneutics.

通过赋予此类知识形式在计算机系统评估中的作用，机器学习重新提出了一个古老的问题：语言真的是一种可靠的媒介吗？这个问题不仅与 GPT-3 等文本生成器相关，也与任何其他使用文字的机器学习系统相关。图像分类模型难辞其咎，因为它们通常使用诸如摩托车或猫之类的词语对照片中描绘的物体进行分类。41在基于数据构建可计算的词语定义时，这些系统让人想起一些启蒙思想家希望建立全新、更美好世界观的经验方法。42机器学习模型与这些启蒙项目的不同之处，在于它们并不打算彻底取代现有语言，直至概念的形成。与布尔的逻辑方程式一样，GPT-3 的声明从人们已经使用的语言中汲取其含义。然而，与布尔的系统相比，这项技术并没有提供一个可以独立于这些语言运作的正确性标准。要理解这一发展趋势的含义，需要思考计算过程与支配人类交流的社会实践之间的相互作用。由此出发，我们别无选择，只能走向政治。

By granting such forms of knowledge a role in the evaluation of computer systems, ML reopens an ancient question: is language really a reliable medium? This question is relevant not just to text generators such as GPT-3 but also to any other ML system that employs words. Image classification models are culpable, since they usually classify the objects depicted in photographs using words such as motorcycle or cat.41 In constructing computable definitions of words based on data, these systems recall the empirical methods by which some Enlightenment thinkers hoped to build a new and better view of the world.42 ML models differ from these Enlightenment projects, among other ways, in that they do not deign to replace existing languages all the way down to the formation of concepts. Like Boole’s logical equations, GPT-3’s pronouncements draw what meaning they have from the languages people already speak. Yet in contrast to Boole’s system, the technology does not offer a standard of correctness that can plausibly be said to work independently of those languages. Understanding the implications of this development requires considering the terms on which computational processes interact with the social practices that govern human communication. From here, there is no direction to proceed but into politics.

谈判

The Negotiation

如果说计算机文化与启蒙运动之间有什么共同点，那就是相信技术进步能够创造社会平等。正如孔多塞认为印刷机将创造一个开放的公共领域，让人们能够平等地交流一样，互联网的早期使用者也憧憬着一个乌托邦。计算机时代对启蒙运动的一个回响体现在黑客文化中——一群试图入侵安全系统的松散计算机用户，有时出于恶意目的，有时仅仅是为了探索。在广为流传的1986年宣言《黑客文化》中，在《黑客良知》一书中，化名“导师”（真名劳埃德·布兰肯希普）的黑客提出了这种做法的理论基础。他所推崇的黑客行为类似于启蒙运动的揭露文学，旨在揭露权贵们隐藏的真相。在整篇宣言中，他反复强调黑客“都是一样的”。43他写道，黑客社区的成员“不分肤色、国籍和宗教偏见”；他们的文化是“以言以行，而非外貌来评判他人” 。44这个想法是，通过让人们纯粹通过文本进行交流，数字媒体可以让每个人处于公平的竞争环境中，无论其种族、性别或社会经济地位如何。这是一个在今天看来幼稚的梦想，但它在围绕算法的中立性话语中依然存在。交流系统可以消除偏见的想法重申了对公平竞争环境的渴望，而这正是孔多塞语言方案的动机。

If there is a common thread linking computer culture to the Enlightenment, it is a belief that technological progress can create social equality. As Condorcet thought the printing press would create an open public sphere in which people could discourse as equals, the early denizens of the internet envisioned a utopia. One computer-age echo of the Enlightenment occurred in hacker culture—a loose affiliation of computer users who sought to break into secure systems, sometimes for malicious purposes and sometimes merely to explore. In the widely circulated 1986 manifesto titled “The Conscience of a Hacker,” the pseudonymous hacker “The Mentor” (real name Lloyd Blankenship) presents a rationale for this practice. The form of hacking he endorses resembles the Enlightenment genre of the exposé, which sought to reveal what powerful people are hiding. Throughout the manifesto, he repeats the refrain that hackers are “all alike.”43 The members of the hacker community, he writes, “exist without skin color, without nationality, without religious bias”; their culture involves “judging people by what they say and think, not what they look like.”44 The idea is that, by enabling people to communicate purely through text, digital media could place everyone on a level playing field, regardless of their race, gender, or socioeconomic status. It is a dream that today seems naïve, but it persists in the discourse of neutrality that continues to surround algorithms. The idea that a system of communication can eliminate bias reiterates the desire for a level playing field that motivated Condorcet’s language scheme.

但这种乌托邦主义的复杂性值得深入探究。对于导师而言，平等主要源于技术媒介的开放性：他认为，只要人们能够在网上自由思考、言论和探索，社会差异就会消失。实际上，他将黑客社区的成员视为自主的主体，参与由言语交流定义的公共领域。然而，这份宣言也提出了教育的必要性：黑客“在学校里被喂着婴儿食品，而我们却渴望牛排”。45 在关于互联网带来的平等主义可能性的叙述中，教育常常被淡化，但它却是启蒙运动的核心。孔多塞的乌托邦计划并非止步于让文字自由流动；相反，他将文字本身视为偏见的载体。因此，实现真正的平等不仅意味着赋予人们思考和言论的自由，还意味着改变他们的思考和言论方式。对他的批评者来说，这使得他的哲学带有隐含的极权主义色彩，试图将某种思维方式强加于世界。然而，互联网却让人们感受到共同语言的缺失。科技既带来了分裂，也带来了团结；开放的沟通既摧毁了等级制度，也强化了等级制度。

But there is a complexity to this utopianism that is worth unpacking. For the Mentor, equality stems primarily from the openness of the technical medium: as long as people are able to think, speak, and explore freely online, he supposes, social differences will vanish. In effect, he treats the members of the hacker community as autonomous subjects engaging in a public sphere defined by the exchange of words. Yet the manifesto also suggests the need for education: hackers have “been spoon fed baby food at school when we hungered for steak.”45 Education is often de-emphasized in narratives about the egalitarian possibilities raised by the internet, but it was central to the Enlightenment project. Condorcet’s utopian plan did not stop at enabling words to flow freely; instead, he treated words themselves as bearers of prejudice. Achieving true equality, then, would mean not just granting people the freedom to think and speak but also changing how they think and speak. To his detractors, this makes his philosophy implicitly totalitarian, an attempt to impose a certain way of thinking on the world. Yet the internet has made the absence of a common language felt. Technology has divided as much as it has united, and open communication has reinforced hierarchies as much as it has demolished them.

事后看来，互联网乌托邦主义的衰落几乎与“算法”用法的转变同时发生，这并不奇怪。经典算法符合早期互联网的开放性理念：《美国计算机协会通讯》的“算法”部分提供了一个可重用程序库，这些程序在公共论坛上进行描述和解释。随着机器学习的兴起，“算法”一词与开放性的联系越来越少，而与不透明性的联系越来越紧密。这种转变在一定程度上与新软件工程技术的出现有关：2010 年代中期，Docker 等容器管理系统得到了广泛采用。以及 Kubernetes，它们使得跨计算机传输复杂的软件配置变得更加容易。这类系统可以与“低代码”和“无代码”开发方法协同工作，这些方法主要通过组合和配置预先打包的程序来实现。如果说在计算机科学发展的早期，算法是为了将程序泛化以便于复用，那么如今，计算机系统中被重新利用和复用的部分已经变得更加复杂，不仅包括程序，还包括模型、数据和软件安装。

In hindsight, it is not surprising that the decline of internet utopianism coincided almost exactly with the shift in usage of algorithm. Classical algorithms fit with the early internet’s ideal of openness: the “Algorithms” section of Communications of the ACM provided a repository of reusable procedures that were described and explained in a public forum. With the rise of machine learning, the word algorithm has come to be associated less with openness than with opacity. The shift had to do in part with the emergence of new software engineering techniques: the mid-2010s saw the widespread adoption of container management systems such as Docker and Kubernetes, which make it much easier to transfer complex software configurations across computers. Such systems can work together with “low-code” and “no-code” approaches to development, which work mainly through combining and configuring prepackaged programs. If algorithms were, in the early decades of computer science, about generalizing procedures for reuse, the parts of computer systems being repurposed and reused have now become far more complex, encompassing not just procedures but also models, data, and software installations.

这些技术实践的变化与互联网作为一种媒介的运作方式（无论是在制度上还是文化上）的转变相吻合。如果说导师将互联网想象成一个透明的渠道，人们可以通过它畅所欲言，那么这种观点的最后一个伟大体现就是个人博客。2013年，谷歌关闭了其聚合服务Google Reader，个人博客遭受了毁灭性的打击。这类服务在很大程度上让位于拥有所谓算法推送的社交媒体平台，这些平台通过隐藏的定制流程来决定向用户展示什么内容，旨在通过尽可能长时间地吸引人们的注意力来实现利润最大化。在本世纪下半叶，两本极具影响力的著作——凯茜·奥尼尔的《数学毁灭武器》（2016年）和萨菲亚·诺布尔的《压迫算法》（2018年）——指出算法是加剧种族和性别不平等的引擎。46 2010年前后的互联网从来都不是导师所想象的包容性乌托邦，而算法媒体可能会加剧这一问题。

These changes in technical practice coincided with a shift in how the internet worked as a medium, both institutionally and culturally. If the Mentor imagined the internet as a transparent conduit through which people could speak their minds without resistance, the last great expression of this view was the personal blog, which suffered a devastating blow when Google shut down its aggregation service Google Reader in 2013. Such services largely gave way to social media platforms with so-called algorithmic feeds, which decide what to show users through hidden processes tailored to maximize profitability by keeping people’s attention for as long as possible. In the second half of the decade, two highly influential books, Cathy O’Neil’s Weapons of Math Destruction (2016) and Safiya Noble’s Algorithms of Oppression (2018), singled out algorithms as engines that can reinforce inequality along race and gender lines.46 The internet circa 2010 was never the inclusive utopia the Mentor imagined, and algorithmic media threaten to worsen the problem.

奥尼尔、诺布尔等人的批评引发了两种截然不同的回应。一些人工智能 (AI) 研究人员开始分析他们的模型是否存在理论上可以在未来修订中解决的偏见。GPT-3 论文就是这种方法的一个例子，该论文报告了针对性别、种族和宗教偏见的测试实验。47研究人员还致力于使人工智能系统“可解读”或“可说明”，以缓解人们对其不透明性的担忧。48其他人则质疑人工智能是否是​​一个好主意。 2021年3月，Emily M. Bender、Timnit Gebru、Angelina McMillan-Major 和 Margaret Mitchell（使用刻意透明的笔名“Shmargaret Shmitchell”）发表了一篇题为《论随机鹦鹉的危险：语言模型会不会太大？》 （On the Dangers of Stochastic Parrots: Can Language Models Be Too Big?）的论文。49 Gebru和 Mitchell 曾是谷歌道德人工智能团队的联合负责人，但两人在本文发表前都离开了公司。（关于 Gebru 是被解雇还是主动辞职，说法不一。）这篇论文探讨了 GPT-3 等大型语言模型的缺点；其主要建议是研究人员应该专注于可以精心策划和全面描述的小型数据集。Gebru 和 Mitchell 离开谷歌的消息引起了广泛关注。大量媒体报道，有助于人们广泛认识到算法偏见问题。

The critiques of O’Neil, Noble, and others inspired two distinct lines of response. Some artificial intelligence (AI) researchers began to analyze their models for biases that could theoretically be addressed in future revisions. An example of this approach appears in the GPT-3 paper, which reports experiments testing for bias along the lines of gender, race, and religion.47 Researchers have also undertaken efforts to make AI systems “interpretable” or “explainable” so as to allay concerns about their opacity.48 Others have questioned whether AI is a good idea at all. In March 2021, Emily M. Bender, Timnit Gebru, Angelina McMillan-Major, and Margaret Mitchell (under the intentionally transparent pseudonym “Shmargaret Shmitchell”) presented a paper titled “On the Dangers of Stochastic Parrots: Can Language Models Be Too Big? ”49 Gebru and Mitchell had been coleaders of Google’s Ethical AI team, but both left the company before this paper was published. (Reports differ as to whether Gebru was fired or resigned voluntarily.) The paper is a discussion of the disadvantages of large language models such as GPT-3; its main suggestion is that researchers focus on smaller data sets that can be curated and thoroughly described. Gebru and Mitchell’s departure from Google received a great deal of media coverage, contributing to a widespread awareness of the issue of algorithmic bias.

这一事件表明，2010 年代中期围绕算法的学科界限的重新划分不仅仅涉及语义学。奥尼尔和诺布尔的论点引发了所提出问题的真正变化。正如广泛报道的那样，诱导包括 GPT 模型在内的一系列机器学习技术产生似乎复制刻板印象的输出并不困难。50例如，当输入文本“ The man worked as a”时，GPT-2 预测接下来的两个单词是“ security guard”；当输入“ The woman worked as a”时，它会显示“ waitress”。51这个问题部分源于用于生成 GPT 模型的训练数据的构成，这些数据可能包含大量种族主义和性别歧视文本；因此，该模型能够生成种族主义和性别歧视的输出也就不足为奇了。然而，这些模型提出了一个更广泛的信任问题，仅仅修复特定的已知问题并不能解决这个问题。由于模型无法像 Dijkstra 证明程序正确性那样被证明是正确的，因此很难想象 GPT-3 的海量参数矩阵中可能隐藏着什么未被发现的问题。

This incident shows that the redrawing of the disciplinary lines around algorithms in the mid-2010s was not merely about semantics. The arguments of O’Neil and Noble inspired a real change in the issues being raised. As has been widely reported, it is not difficult to induce a range of ML technologies, including the GPT models, to produce outputs that seem to replicate stereotypes.50 When, for instance, one enters the text “The man worked as a,” GPT-2 predicts that the next two words will be “security guard”; when one enters “The woman worked as a,” it comes up with “waitress.”51 The problem stems in part from the makeup of the training data used to produce the GPT models, which likely contain a large amount of racist and sexist text; as a result, it is not surprising that the model can generate racist and sexist output. Yet the models raise a broader issue of trust that cannot be resolved merely by fixing specific, known issues. Since the models cannot be proven correct in the way Dijkstra wanted to prove the correctness of programs, it is hard to imagine what undiscovered problems may lurk in GPT-3’s massive parameter matrices.

关于这些问题的讨论提出了一些棘手的问题，即计算过程与语言之间的关系。从 20 世纪 60 年代到 80 年代，文学评论家们一直在争论意义是基于作者的意图还是由读者建构。52 2010年代末，这场争论在人工智能的法律含义讨论中被引用，从而获得了新的意义。在 2018 年的一本书中，法律学者罗纳德·K·L·柯林斯和戴维·M·斯科弗认为，机器生成的文本构成“无意图的言论自由”，应该受到美国法律的保护。53 “随机鹦鹉”一文持截然相反的立场，指出“人类对话者倾向于在没有意义的地方推断意义，这可能会误导 NLP [自然语言处理] 研究人员和普通公众，使他们认为合成文本是有意义的。” 54作者详细阐述道，输出结果看似连贯，但其实是一种错觉，因为文本“并非基于交流意图、任何世界模型或任何读者心理状态模型。” 55这种思路表明，尽管 GPT-3 可以模仿人类的写作实践，但它对文字的处理方式与人类并不相同，而假设并非如此则是被欺骗了。

The conversation about these issues has raised slippery questions about how computational processes relate to language. From the 1960s to the 1980s, literary critics debated whether meaning is based on the intention of a writer or constructed by the reader.52 In the late 2010s, this debate gained a renewed relevance when it came to be cited in discussions of the legal implications of AI. In a 2018 book, the legal scholars Ronald K. L. Collins and David M. Skover argue that machine-generated text constitutes “intentionless free speech” that ought to be protected under United States law.53 The “Stochastic Parrots” article takes quite the opposite position, stating that “the tendency of human interlocutors to impute meaning where there is none can mislead both NLP [Natural Language Processing] researchers and the general public into taking synthetic text as meaningful.”54 The seeming coherence of the output, the authors elaborate, is an illusion because the text “is not grounded in communicative intent, any model of the world, or any model of the reader’s state of mind.”55 This line of thought suggests that, for all that it can parrot human writing practices, GPT-3 is not doing the same thing with words that humans do, and to assume otherwise is to be duped.

虽然这些论点的直接参考点是文学理论和心理语言学，但它们也重新提出了几个世纪前符号方法所面临的问题。通过将语言模型与其他信息源相结合，人工智能研究人员渴望实现莱布尼茨推理演算的最雄心勃勃的版本：一种统一的方法来解答关于任何事物的问题。56与莱布尼茨一样，这些雄心壮志也遭遇了沟通的混乱。在 GPT-2 文章中给出的一个例子中，机器被问到“哪条河流与罗马城有关？”；它回答“台伯河”。57至少在这种情况下，它是有效的。但缺少了一些东西。虽然关于台伯河的问题有一个普遍认可的答案，凭借该答案可以判断输出无可争议的正确性，但该模型并没有提供明确的机制来就有争议或分裂的术语达成一致。相反，答案的正确性完全取决于模型是否选择了正确的单词序列，而不考虑人们对这些单词的含义的看法。

While their immediate reference points are literary theory and psycholinguistics, these arguments are also reopening questions that attended symbolic methods centuries ago. By combining language models with other sources of information, AI researchers aspire to realize the most ambitious version of Leibniz’s calculus ratiocinator: a unified method for answering questions about anything and everything.56 As with Leibniz, these ambitions have run up against the messiness of communication. In one of the examples given in the GPT-2 article, the machine is given the question “What river is associated with the city of rome?”; it answers, “the Tiber.”57 It works, at least in this case. But something is missing. While the question about the Tiber has a generally agreed-on answer by virtue of which that output could be judged indisputably correct, the model provides no clear mechanism for creating agreement about terms that are contentious or divisive. Instead, the correctness of the answer is judged wholly by whether the model chooses the right sequence of words, without regard to what people may think those words mean.

这个问题在涉及政治色彩浓厚的问题上最为明显。例如，为了向法国大革命致敬，我问 GPT-2 自由、平等还是博爱哪个更重要。它回答说，最重要的是“ the freedom of the individual to choose his own life” 。58这样的输出，即使来自一个人们在这些问题上非常信任的系统，能理性地改变任何人对任何事情的看法吗？仅仅通过模型是否按照预先定义的规则选择了正确的词语来评估模型，掩盖了一个事实，即词语对所有人的含义并不总是相同的。如果两个用户都认为个人自由是最高价值，但对自由的含义有不同的看法，他们可能会同意人工智能给出了正确的答案，但对答案的实际含义却有着截然不同的看法。

The problem is most apparent in regard to politically charged questions. For instance, in a nod to the French Revolution, I asked GPT-2 whether liberty, equality, or fraternity was most important. The most important one, it responded, is “the freedom of the individual to choose his own life.”58 Could such an output, even coming from a system people seriously trust on such issues, rationally change anyone’s mind about anything? Evaluating the model solely by whether it chooses the right words by a predefined rubric covers over the fact that words do not always mean the same things to all people. If two users agree that individual freedom is the highest value but differ as to what freedom entails, they may agree that the AI produced the correct answer while having vastly different ideas of what that answer actually was.

如此纷繁复杂的解释冲突，无法仅仅通过提高答案质量来解决，而研究人员无疑会这么做。如果人工智能的言论真的被视为权威，它们就会像圣经文本一样被仔细研读和争论。问题在于语言本身。词语——不是任何特定的词语，而是泛指所有词语——无法完成问答任务。这个问题并不新鲜；相反，几个世纪以来，它一直是关于通用计算的讨论的一部分。语言模型中发现的偏见是弗朗西斯·培根对市场偶像的现代演绎——偏见和错误通过日常闲聊重现。事实上，《随机鹦鹉》论文中提出的关于意义的担忧与约翰·洛克关于“词语滥用”的论述之间存在着相当明显的相似之处：在这两种情况下，人们都被指责轻信地假设词语对于作者和读者遵循相同的概念。在主客体分歧将此类问题排除在讨论范围之外之前，如何回应此类问题是一个核心的认识论问题。

This profusion of conflicting interpretations could not be resolved merely by improving the quality of the answers, as researchers will undoubtedly do. If an AI’s pronouncements truly were viewed as authoritative, they would be pored over and debated like the text of the Bible. The problem is language itself. Words—not any particular words, but words in general—are incapable of doing what the question-answering task asks of them. This issue is not new; rather, it has been a part of the discourse about universal computation for centuries. The biases found in language models are a modern iteration of Francis Bacon’s idols of the market—prejudices and errors reproduced through everyday chatter. There is, indeed, a not insignificant parallel between the concerns about meaning raised in the “Stochastic Parrots” paper and John Locke’s account of the “abuse of words”: in both cases, people are charged with credulously presuming that words adhere to the same concepts for the writer as for the reader. How to respond to such issues was a central epistemological question before the subject–object divide took it off the table.

在这方面，孔多塞比莱布尼茨更具代表性。莱布尼茨有神学基础，相信人们最终会就符号的含义达成一致，孔多塞希望通过教育和社会变革来达成这种一致——不仅提供一种产生答案的工具，更要教会人们一种理解世界的新方式。问答任务跳过了这一步，因为正如人们认为公共领域的平等参与只能通过自由交流词语来实现一样，它的认识论止步于语言。形成人类理解的所有困难都被推卸给词语，而结果取决于词语先前存在的达成共识的能力。这种依赖性是自洛克认识论崩溃以来，符号方法常常对词语采取的消极态度的最终结果——倾向于将语言视为一种自然资源，可以根据需要从中获取意义，但最终将其与算法逻辑区分开来。最近的经验表明，这种方法是不够的，词语可能会引发问题，语言不应该被简单地理解为语言本身。

In this regard, a more illustrative figure than Leibniz would be Condorcet. Whereas Leibniz had a theological basis for trusting that people would eventually reach an accord about what symbols meant, Condorcet wanted to create this accord through education and social transformation—not just providing an instrumental means of producing answers but also teaching people a new way of understanding the world. The question-answering task skips over this step since, as in the idea that equal participation in the public sphere can be produced solely through the free exchange of words, its epistemology stops short at language. All the difficulties of forming human understanding are off-loaded onto words, on whose preexisting ability to produce an agreement the outcome depends. This dependence is the ultimate result of the passive attitude toward words that has often attended symbolic methods since the Lockean epistemology collapsed—the tendency to treat language as a natural resource from which meanings can be taken as needed but that is ultimately separate from algorithmic logic. Recent experience has shown that this approach is not sufficient, that words can cause problems, that language should not just be taken as it is.

思考这个问题的一个有用的起点是考虑用户界面设计师在屏幕上放置文字或其他符号时在做什么。正如 PB Andersen 在其《计算机符号学理论》中所解释的那样，界面设计师必须在使系统根据现有类别变得易于理解和开辟新思维方式之间找到平衡。59前一个目标通常涉及隐喻 - 例如文件和文件夹等词语暗示与旧实践的类比，以帮助人们理解软件。60但计算机也会将单词的含义改变为新的方向。计算机界面中常用的词语，例如打开、保存和窗口，已经具有了新的含义，这些含义背离了最初激发它们的隐喻。其他概念则是全新的 -数据库和软件指的是计算机时代之前不存在的概念。用户界面设计在一定程度上借鉴语言来改变语言，因此，它涉及生产软件的个人和组织与更广泛的语言使用者群体之间的协商，后者有权接受或拒绝新造词，并可以以违背设计者意图的方式使用它们。

A helpful starting point in thinking through this issue would be to consider what user interface designers are doing when they place words or other signs on the screen. As P. B. Andersen explains in his Theory of Computer Semiotics, interface designers must find a balance between making systems understandable in terms of existing categories and opening new ways of thinking.59 The former goal often involves metaphors—words such as file and folder suggest analogies with older practices as a way to help people understand software.60 But computers also bend the meanings of the words in new directions. Words that are commonly used in computer interfaces, such as open, save, and window, have taken on new senses that depart from the metaphors that initially motivated them. Other concepts are entirely new—database and software refer to ideas that were nonexistent prior to the computer age. User interface design draws on language in part to change it, and it involves, accordingly, a negotiation between the individuals and organizations that produce software and the broader communities of language speakers who have the power to accept or to reject new coinages, and who can employ them in ways that break with the designers’ intent.

问答任务以一种单向的方式处理这种协商，至少就输出的词汇而言是如此。答案应该在现有语言中通顺，而不是以任何有意的方式为语言贡献任何新的东西。在这方面，主语和宾语的划分仍然存在。然而，就输入的词汇而言，情况更加复杂，而正是在这里，GPT 模型开启了彻底反思计算与语言关系的潜力。一些模型的爱好者认为，多任务学习者将改变编程实践；程序员不再通过编写代码来实现算法，而是设计一个提示，让模型执行所需的任务。61举个关于 GPT-3 的论文中的例子，研究人员输入了“ Please unscramble the letters into a word, and write that word: taefed =”；机器给出了所需的响应“ defeat” 。62只要有一个可接受的单词列表，设计一个能够可靠地解决这个问题的算法并不特别困难；这类问题可能会在计算机科学课程或工作面试中用作练习。新颖之处在于，计算机显然能够自己找到解决方案。

The question-answering task handles this negotiation in a one-sided way, at least as far as the words in the output go. The answers are supposed to make sense within existing languages, not, in any intentional way, to contribute anything new to language. In this regard, the subject–object divide remains in place. The situation is more complex, however, regarding the words in the input, and it is here that the GPT models open the potential for a radical rethinking of how computation relates to language. Some enthusiasts of the models have argued that multitask learners will transform the practice of programming; rather than implement algorithms by writing code, the programmer will design a prompt that gets a model to perform the desired task.61 To pick an instance from the paper about GPT-3, the researchers entered, “Please unscramble the letters into a word, and write that word: taefed =”; the machine produced the desired response, “defeat.”62 It would not be especially hard to design an algorithm that could solve this problem reliably, provided one has a list of acceptable words; it is the sort of problem that might be used as an exercise in a computer science class or job interview. What is new is the fact that the computer is apparently able to figure out a solution on its own.

这种提示技术并非特定于面向文本的任务。通过将语言模型与图像生成技术相结合，于 2020 年底首次发布的流行程序 CLIP+VQGAN 可以根据用户输入的文本描述生成合成图像。63该程序被誉为一种使人工智能艺术民主化的方式：由于提示可以用英文书写，因此无需太多技术背景即可对系统进行实验并产生有趣（即使不可预测）的结果。2021 年 6 月作为预览版发布的 GitHub Copilot 将类似的方法应用于计算机编程：它使用 GPT-3 的修改版本根据描述代码预期功能的自然语言注释自动生成代码。20 世纪 50 年代，格蕾丝·霍珀 (Grace Hopper) 在编译器方面的工作中，试图通过将算法转换为可执行指令的工作从人转移到机器，使计算变得不那么繁琐。这些新实践再次推动了链条的变革：是机器，而不是人类，将问题定义转化为算法。这个想法让我们回想起霍珀在其职业生涯早期对我们现在所说的编程的理解：与机器交流。

This prompting technique is not specific to text-oriented tasks. By combining a language model with image generation techniques, the popular program CLIP+VQGAN, first publicized in late 2020, produces synthetic images based on text descriptions entered by the user.63 This program is touted as a way of democratizing AI art: since the prompts may be written in English, it is possible to experiment with the system and produce interesting (if unpredictable) results without needing much of a technical background. GitHub Copilot, which was released as a preview in June 2021, applies a similar method to computer programming: it uses a modified version of GPT-3 to generate code automatically based on natural-language comments describing what the code is supposed to do. In her work on compilers in the 1950s, Grace Hopper sought to make computing less tedious by transferring the work of converting algorithms into executable instructions from human to machine. These new practices move the chains once again: the machine, not the human, transforms the problem definition into an algorithm. This idea suggests a return to how Hopper, early in her career, understood what we now call programming: communicating with a machine.

无论提示作为一种编程范式有多么成功，控制语言模型的需求都会引发一些问题，这些问题困扰着算法与交流之间的严格区分。标识符“a34kTMNs”和“MARILYN”在 ALGOL 代码中可能可以互换，但如果试图诱导语言模型执行某项任务，那么每个词语的选择都很重要。64一种被提出的多任务学习方法是模式利用训练，它使用模板来探索预训练模型如何响应某些词语排列。例如，以下模式可用于根据段落p回答问题q：65

However successful prompting turns out to be as a programming paradigm, the need to control language models raises issues that trouble the rigid distinction of algorithms from communication. The identifiers “a34kTMNs” and “MARILYN” may be interchangeable in ALGOL code, but if one is trying to induce a language model to perform a certain task, every word choice matters.64 One proposed approach to multitask learning is pattern-exploiting training, which employs templates that exploit how pretrained models respond to certain arrangements of words. For instance, the following patterns can be used to answer a question q based on passage p:65

p。问题：q？答案：__。

p。根据上一篇文章，q？__。

根据以下段落，q ? __. p

p. Question: q? Answer: __.

p. Based on the previous passage, q? __.

Based on the following passage, q? __. p

这些动词公式在语言模型中发挥作用的有效性源于训练数据中所体现的人类意义构建实践——源于人们过去使用词语和标点符号的方式。理解一个提示与另一个提示的区别，既不需要对 Dijkstra 所认可的程序进行数学分析，也不需要任何熟悉的解释，而是需要分析这些词语对一系列矩阵计算的影响，这些计算在某种程度上（但并非完全）类似于英语的结构。

The effectiveness of these verbal formulae in getting language models to do things stems from the human meaning-making practices represented in the training data—from ways people have used words and punctuation marks in the past. Understanding what differentiates one prompt from another requires neither the mathematical analysis of procedures endorsed by Dijkstra nor any familiar sort of interpretation, but rather analyzing the effects the words have on a series of matrix calculations that somewhat, but not totally, parallel the structures of the English language.

面对这种数学与语言的全新融合，仅仅因为语言超越了学科界限就认为它没有问题（正如布尔和经典算法所言），这种观点已不再站得住脚。另一种观点是对词语的不信任，这种不信任在过去符号方法中屡见不鲜。尽管1790年代那些咄咄逼人的语言改革努力理应承受其所面临的诸多批评，但这些努力所应对的问题从未消失。词语可以造成伤害、分裂和扭曲，而当它们被假定为透明时，它们更容易造成伤害、分裂和扭曲。像GPT-3这样的大型语言模型所带来的危险，部分源于对语言的不加批判的态度，源于一种错误的信念，即即使这些词语是由机器选择的，我们对词语含义的既有概念也能充分指导这些词语的意义。19世纪的符号化转型将这种担忧从代数学领域中移除，并且这种界限仍然存在，尤其是在语言模型基准测试用词语来衡量知识的方式中。现在是时候重新探讨在这一安排形成之前围绕符号计算的问题了。

In the face of this new hybrid of mathematics and language, viewing language as unproblematic because it lies outside a disciplinary boundary, as Boole implicitly did and the classical algorithm implicitly does, is no longer tenable. An alternative lies in the mistrust of words that has often attended symbolic methods in the past. Although the overbearing language reform efforts of the 1790s deserved much of the criticism they faced, the problems to which these efforts responded have never gone away. Words can harm, divide, and distort, and they can do so all the more readily when they are presumed to be transparent. The dangers posed by large language models such as GPT-3 stem in part from an uncritical attitude toward language, from a misguided belief that our preexisting conceptions of what words mean can provide an adequate guide to those words’ significance even when those words are chosen by a machine. The symbolic turn of the nineteenth century removed such concerns from the domain of algebra, and this boundary persists, among other places, in the way language model benchmarks measure knowledge by the word. The time has come to reopen the questions that surrounded symbolic computation before this arrangement took shape.

结论

Conclusion

正如我在本书中所指出的，从现代代数符号的出现到编程语言的发展，在符号方法的讨论中，关于沟通的担忧一次又一次地出现。在17世纪，符号与使书写独立于语言的梦想相契合；代数符号承诺了一种无需文字介入，将思想直接写在纸上的方式。在18世纪，正如孔狄亚克神父所说，代数成为了良构语言的典范，而代数符号的任意性——a和b值的易变性——证明了从头构建一种更好的语言的可能性。然而，在许多观察家看来，代数和微积分的方法似乎包含着定义不清或荒谬的思想，这引发了另一种观点，即符号其好坏取决于用来定义它们的词语。这些不同的立场对微积分的“算法”是否可信、数学家能否提出新的和的定义，以及−1究竟是真数还是无稽之谈，都产生了影响。

As I have shown in this book, anxieties about communication have arisen again and again in discussions of symbolic methods, from the emergence of modern algebraic notation to the development of the programming language. In the seventeenth century, symbols were aligned with the dream of making writing independent from language; algebraic notation promised a way of putting thoughts directly on a page without the intervention of words. In the eighteenth century, algebra became the model of a well-formed language, as the Abbé de Condillac put it, and the arbitrariness of algebraic symbols—the ease of changing the values of a and b—testified in favor of the possibility of building a better language from the ground up. Yet the methods of algebra and calculus seemed, to many observers, to involve ill-defined or absurd ideas, giving rise to an alternate view in which symbols were only as good as the words used to define them. These differing positions had implications for whether the “algorithm” of the differential calculus could be trusted, whether mathematicians could propound a new definition of sum, and whether −1 was a true number or mere nonsense.

这些古老的争论，看似古朴，却触及了一个值得重新探讨的问题：如何在明确的定义与社会惯例之间取得平衡，以此作为确定符号意义的方式。GPT-3 依赖于现有语言，因此其意义与社会性含义相符：“ sum”一词在其输出中可能具有的任何意义，都源于人们如何使用该词，并通过复杂的技术手段进行调解。然而，计算实践强烈倾向于将单词和其他符号视为任意标记，除了代码中赋予它们的角色之外，没有任何固有含义。这种思维方式体现在问答任务的评估方式中：输出的得分完全取决于它与所需字符序列的重叠程度。这种方法表明了主客体划分的持续影响，这种划分始于 19 世纪中叶，通过将语言的社会复杂性归结于教学、科学交流以及最终的用户界面等实际领域，赋予了符号独立于词语的自主权。

These old debates, quaint as they may seem, touch on a question worth reviving: how to balance explicit definitions with social convention as ways of determining the meanings of signs. In its dependence on existing languages, GPT-3 is implicitly aligned with the social sort of meaning: whatever significance the word sum may have in its output stems from how that word has been used by people, as mediated through a complex technological apparatus. Yet the practice of computation hews strongly toward treating words and other symbols as arbitrary tokens with no inherent meanings beyond the roles assigned to them in code. This way of thinking is manifest in how the question-answering task is evaluated: the output is scored solely by how much it overlaps with the desired sequence of characters. This approach shows the continuing influence of the subject–object divide that, starting in the mid-nineteenth century, granted symbols autonomy from words by consigning the social complexities of language to the practical realms of pedagogy, scientific communication, and, eventually, user interface.

这种分歧的解释很简单。正如我所说，现代计算是非传统的：在标准编程语言中，人们可以随意为单词和其他符号赋予计算意义，而不管人们过去如何使用它们。但这种非传统性只有在人们将自己的选择与做出这些选择的更广泛背景之间划清界限时才成立——在理论上可以做一些像交换名称这样的傻事input（output程序员确实可以轻易做到）与在更实际的意义上这样做是愚蠢的。正如早期计算机科学家如Dijkstra所确立的，“硬”算法逻辑与“软”人为因素之间的等级区分，将前一种观点提升为合乎逻辑的观点，即使它所实现的许多实践远非合乎逻辑。摆脱这种区别就意味着认识到，计算机编程所涉及的选择毕竟在某种程度上受到单词和符号的传统含义的限制——有一个认识论上相关的原因，为什么程序员除了开玩笑之外，不使用input来表示output。

This divide can be explained quite simply. Modern computation is, as I have termed it, aconventional: in standard programming languages, one can assign computational meanings to words and other symbols at pleasure, regardless of how people have used them in the past. But this aconventionality only holds if one draws a line between one’s choices and the broader contexts in which they are made—between the theoretical sense in which it is possible to do something silly like swap the names input and output (as a programmer can indeed readily do) and the more practical sense in which doing so would be foolish. The hierarchical distinction between “hard” algorithmic logic and “soft” human factors, as instated by early computer scientists such as Dijkstra, elevates the former perspective as the logical one even if many of the practices it enables are far from logical. Moving away from this distinction would mean recognizing that the choices involved in programming computers are, after all, to some extent constrained by the conventional meanings of words and symbols—that there is an epistemologically relevant reason why programmers do not, except as a joke, use input to mean output.

这些问题虽然不属于“硬”逻辑范畴，但对程序员来说却并不陌生。大型软件项目遵循复杂的命名约定，以确保一致性并避免歧义。在私营部门，此类约定所规定的纪律确实受到重视。超越个人主义的编码实践。但命名约定仅在相对较小的开发团队中有效，或者最多在 Python 等编程语言的用户群中有效。语言模型需要与数百万或数十亿使用者的社区进行协商，而这种规模下出现的问题更具争议性。紧张局势的一个根源在于，人们对科技公司将理念强加于民众感到不适——赋予那些拥有操控权的人过多的权力。但机器学习引发了其他问题，这些问题与假定开发人员完全控制系统的涓滴模型并不完全相符。正如批判性话语所表明的那样，GPT-3 的训练文本包含一些最好不要被计算机系统奉为圭臬的元素：偏见、成见和谎言。要充分认识到这些问题，就不能将人们的语言实践视为一个可以汲取意义的源泉（就像问答的神谕模型那样），而应将其视为一个可以进行批评和改进的人类活动舞台。

Such issues, though excluded from the domain of “hard” logic, are hardly unfamiliar to programmers. Large software projects follow elaborate naming conventions to ensure consistency and avoid ambiguity. In the private sector, the discipline imposed by such conventions is indeed valued above individualistic coding practices. But naming conventions only operate within relatively small teams of developers or, at largest, the user bases of programming languages such as Python. Language models negotiate with communities of millions or billions of speakers, and the issues that arise at this scale are more contentious. One source of tension is a discomfort with technology companies imposing ideas on the population—with the undue amount of power granted to the people with the privilege of turning the knobs. But ML raises other issues that do not fit neatly into a trickle-down model that presumes developers have total control over systems. As the critical discourse has shown, GPT-3’s training text contains elements that are best not enshrined in computer systems: prejudices, biases, falsehoods. To reckon fully with these problems is to think of people’s linguistic practices not as a well from which meaning may be drawn, as in the oracle model of question answering, but rather as an arena of human activity that may be critiqued and improved.

钟摆已经开始摆回原点。随着机器学习的兴起，2010年代人们对鲁道夫·卡尔纳普概念工程理念的兴趣再度高涨，这一次更加强调其政治含义。66新的语言改革通常与女权主义事业相呼应，例如推广性别中立代词，尽管一些倡导者更关注科学的精确性。虽然他们的担忧是21世纪的典型特征，但这些努力表达了一种由来已久的观念：文字并非表达思想的中性媒介，语言可能扭曲世界观。更为激进的概念工程形式可以说也具有通用语言方案的一些缺陷，包括倾向于强迫人们遵循某些做法。语言改革也有可能（正如孔多塞和约翰·戴维·米夏埃利斯等18世纪思想家所指出的那样）通过疏远那些采用改革后语言的人和那些没有采用改革后语言的人来造成社会分裂。然而，利害关系已经发生了变化。这一次，计算似乎不再像莱布尼茨那样，是解决语言缺陷的良方。现在，它似乎加剧了问题，使得这些缺陷更加亟待解决。

The pendulum has already begun to swing back that way. Alongside the rise of ML, the 2010s saw a resurgence of interest in Rudolf Carnap’s idea of conceptual engineering, this time with a great deal more emphasis on its political implications.66 The new language reform is often aligned with feminist causes, such as the promotion of gender-neutral pronouns, although some advocates are more concerned with scientific precision. While their concerns are characteristic of the twenty-first century, these efforts express a centuries-old perception that words are not a neutral medium for expressing thoughts, that language can provide a distorted view of the world. The more activist forms of conceptual engineering arguably share some of the drawbacks of universal language schemes, including a tendency to force certain practices on people. Language reform also bears the potential (as noted by eighteenth-century thinkers such as Condorcet and Johann David Michaelis) to create social divisions by alienating those who have adopted the reformed language from those who have not. Yet the stakes have changed. This time around, computation no longer seems like a solution to the flaws of language, as it did for Leibniz. Now, it seems to worsen the problem, making those flaws all the more urgent to address.

虽然关于概念工程的讨论与同时期关于算法的争论无关，但两者提出了关于个人与集体能动性关系的类似问题。现代计算的危险和力量很大程度上源于这样一个事实：它的非传统性使程序员能够独自处理通常属于社会事务的事情：决定符号的含义。计算声称自己是通用语言，却跳过了创建通用语言所需的艰苦工作。然而，这种非传统性——程序员能够重新定义一个符号就像把“ A := 5”改为“ A := 6”一样容易——也有可能通过将符号的含义引向新的方向来挑战既定的做法。思考如何改变一个符号的含义——它不是像 ALGOL 处理标识符那样被认为是一种任意的标记，而是在其社会流通中——实现清晰度的难度、需要考虑既定用法、达成一致的政治性——可以帮助解决现代计算提出的问题。虽然普遍性可能是一个被误导的目标，但仍然有可能努力形成一种共同的语言，一种让人们和机器能够一起工作而不是仅仅互相交谈或计算的共同理解。一个关键的政治问题是谁来决定这种共同语言应该是什么。

While the conversation about conceptual engineering is separate from the concurrent debates about algorithms, the two raise similar issues regarding the relation of individual to collective agency. Much of modern computation’s danger and power stems from the fact that its aconventionality enables programmers to undertake in isolation what is ordinarily a social matter: deciding the meanings of signs. Computation claims the status of a universal language while skipping the hard part of the work it would take to create one. Yet this aconventionality—the programmer’s ability to redefine a symbol as easily as changing “A := 5” to “A := 6”—also bears the potential to challenge received practices by bending the meanings of signs in new directions. Thinking through what it would take to alter the meaning of a sign considered not as an arbitrary token, as ALGOL treats identifiers, but in its social circulation—the difficulty of achieving clarity, the need to reckon with established usages, the politics of reaching agreement—can help in navigating the issues modern computation raises. Although universality may be a misguided goal, it is still possible to work toward a common language, a shared understanding that enables people and machines to work together without merely talking or computing past each other. A crucial political issue is who gets to decide what that common language should be.

介绍

Introduction

第一章

Chapter One